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# How does the length of a single slit aperture in a ripple tank affect the angle of diffraction generated after plane water waves pass through the single slit?

## Introduction

When studying the topic of waves in physics class, I was keen to understand how something intangible and sometimes invisible could have such major effects on daily life. When I came across the topic of wave behavior, I was determined to clear some of my doubts by studying how natural phenomena, like wave refraction or diffraction, could explain how some of the things I use on a daily basis, such as the internet, work.

This investigation is rooted in my desire to better understand the phenomenon of wave diffraction. When I first studied wave diffraction in physics class, I was confused and fascinated at the same time. I wondered why waves bent after moving across an obstacle or passing through an aperture, and whether there was a way to quantify this bending. I also pondered upon how the diffraction pattern depended on specific wave characteristics, such as frequency or wavelength, and on the size of the obstacle or aperture that causes the wave to diffract.

In this investigation, I will intend to explore the relationship between the length of a single slit aperture and the overall diffraction pattern generated after plane water waves pass through a single slit in a ripple tank. In particular, this research will aim to quantify how the diffraction pattern changes with the single slit length on a ripple tank by measuring how the angle of diffraction varies with this independent variable.

## Background Information

Diffraction is a physical phenomenon deeply present in the nature of waves. All types of waves, whether they are electromagnetic or mechanical, behave in a particular way whenever they pass through an aperture or move around an obstacle: they spread out. This phenomenon is known as diffraction: “the bending of a wave around the edges of an opening or an obstacle.” Diffraction is pivotal for the creation of the physical world as we know it; from being able to hear sounds from places that we are not currently at to being able to see our natural environment, diffraction plays a crucial role in enabling all this to happen.

A well-known diffraction pattern is observed when a wave passes through a single slit aperture. Figure 1 shows a pronounced diffraction pattern for this case. After the wave has passed through the single slit, circular wavefronts are clearly observed. Moreover, the measure of the angle of diffraction is close to $$\frac{\pi}{2}$$ rad, which leads to small shadow regions generated on the sides. Thus, a pronounced diffraction pattern enables a wave to propagate in such a way that its almost perfectly circular wavefronts will reach places that seemed unreachable at a first glance.

Figure 1 - Labelled Diagram Of A Pronounced Single Slit Diffraction Pattern

The extent to which a single slit diffraction pattern is pronounced greatly depends on the relationship between the wavelength (λ) and the length of the single slit. Physical theory explains that a wave’s diffraction pattern is most pronounced when the wavelength is comparable to or slightly greater than the length of the single slit aperture. In this scenario, almost perfectly circular wavefronts, an angle of the diffraction close to $$\frac{\pi}{2}$$ rad, and little-to-no shadow regions on the sides are expected. Conversely, physical theory indicates that, as the length of the single slit aperture increases, thus becoming greater than the wavelength, the diffraction pattern is expected to gradually diminish, which is why circular wavefronts are no longer generated after the diffraction. Instead, linear wavefronts with curved edges are more likely to be observed. This leads to a smaller angle of diffraction, and hence, larger shadow regions on the sides.

Figure 2 - A Comparison Between The Diffraction Patterns Generated By a Small Single Slit And A Large Single Slit

Figure 2 is a real-life photograph of diffraction patterns of water waves in a ripple tank after passing through a small and a large single slit. The previously noted observations can be appreciated in these two pictures. For instance, it is shown that the diffraction pattern generated by the single slit of length comparable to the wavelength leads to almost perfectly circular wavefronts and a large angle of diffraction. The other picture, on the other hand, shows that when water waves pass through a single slit with a length much greater than the wavelength, the generated wavefronts are only curved at the edges, the angle of diffraction is much smaller, and the area of the shadow regions is expected to be greater than that when the single slit is smaller.

Nevertheless, as observed in figure 2, real-life observations of water wave single slit diffraction are not as clear as shown in figure 1. For instance, in the same diffracted wavefront, one can observe parts, such as the edges of the wavefront, where the amplitude is significantly smaller than at the center. This can be explained by the pattern of alternate maxima and minima (generated by constructive and destructive interference amongst the diffracted wavefronts) that is present in single slit wave diffraction, as it indicates that wave intensity, and hence, amplitude is attenuated after the first minima. To limit the scope of this investigation and to improve the accuracy of the results obtained, the start of a shadow region will be determined to be the points, within a diffracted wavefront, that can be said to generate the first minima (any observance of a diffracted wavefront after these points will be ignored and assumed to be part of the ‘shadow region’). Moreover, due to the empirical nature of this investigation, these points will not be assumed to be connected by a perfect diagonal line as in figure 1. Thus, the results obtained will be analyzed within the scope and purview of only having considered diffracted wavefronts until the specific points that we establish generate the first minima pattern.

The aim of this experiment is to investigate the relationship between the length of a single slit on a ripple tank and the angle of diffraction generated after plane water waves pass through the single slit. Special consideration must be given when trying to measure the angle of diffraction since determining clear points that can be said to generate the first minima can be uncertain and ambiguous. For this reason, the angle of diffraction will be measured by plotting specific coordinates, within each diffracted wavefront, at points where there is the first notorious disparity in wave amplitude and intensity between two points. Thus, it is intended that the specific coordinates resemble the points at which the first minima is generated. Then, mathematical software will be used to draw a line of best fit for the plotted coordinates and to calculate the angle between the line of best fit and the y-axis. For simplicity, this investigation will refer to the plotted coordinates as being placed at ‘the end’ of each diffracted wavefront.

It should also be noted that, to improve the accuracy of the measurements, this research will use the strobe light effect. The strobe light effect explains that, whenever a strobe light flashes at the same or at a multiple of a wave’s frequency, the wave’s motion seems to freeze; thus, in this investigation, it will be used to create illusory stationary water waves that will enable much more precise and accurate measurements.

• David McKay

Ivy league graduate, 4+ yrs of experience, vetted IB Examiner!

• ## Hypothesis

In this experiment, the range of the graph of angle of diffraction (θ) against single slit length would be 0 rad $$≤ θ ≤ \frac{π} 2$$ rad, as θ cannot exceed an angle of $$\frac{\pi}{2}$$ rad or go below a 0 rad angle. Moreover, θ is expected to be $$\frac{\pi}{2}$$ rad for a single slit length that is similar to the measurement of the wavelength, as the diffraction pattern is predicted to be most pronounced when the length of the single slit is comparable to the value of the wavelength. The graph is also expected to have a generally negative slope since the angle of diffraction will gradually decrease as the single slit length increases. Additionally, the magnitude of this negative slope is expected to be greater for smaller values of single slit length since, as the length of the single slit increases, the extent to which the wavefronts bend at the edges is not as noticeable as for smaller single slit lengths. Thus, the angle of diffraction is expected to change to a lesser extent for larger single slit lengths, which would lead to a gradual decrease in the magnitude of the negative slope as the single slit length increases.

## Variables

### Independent Variable -

The independent variable of this experiment is the length of the single slit aperture. It will range from 0.4 cm to 2 cm so as to obtain measurements that consider the behavior of the water waves when the single slit aperture is close to the value of both half and double the wavelength (which is 1 ± 0.01 cm for this experiment).

• David McKay

Ivy league graduate, 4+ yrs of experience, vetted IB Examiner!

• ### Dependent Variables -

The dependent variable of this experiment is the angle of diffraction generated after plane water waves pass through the single slit. As previously stated, the angle of diffraction will be measured by first plotting, on graph paper to scale, specific coordinates at points, within each diffracted wavefront, where there is the first notorious disparity in wave amplitude and intensity so as to intend to resemble the points that generate the first minima. Then, mathematical software will be used to draw and calculate the slope of the line of best fit for the plotted coordinates at both the left and right-hand side end of each diffracted wavefront. Finally, to improve the accuracy of the findings, the angle of diffraction will be calculated with the following formulae I have devised -

$$θ = \frac{π}2$$ − arctan(slope)          if slope > 0

$$\theta=arctan(slope)+\frac{\pi}2$$          if slope < 0

### Controlled Variables -

The wavelength of the water waves (1.00 ± 0.01 cm) will be controlled by not changing the frequency at which the ripple motor vibrates(thus the frequency of water waves will also remain constant). It is pivotalto controlthis variable as a change in the wavelength would affect the overall diffraction pattern after the water waves pass through the single slit.

The water depth will be controlled by pouring a specific amount of water (115 ± 5 ml) in the tank at the start of the experiment and by preventing water from dripping throughout the experiment. Moreover, in order to avoid that the metal plates that will act as wave barriers generate a fluctuation in the water depth, they will be placed inside the tank before pouring water into it. Thus, the change in water depth will be almost negligible when moving the metal plates to change the length of the single slit aperture. It is important to control this variable as a change in water depth could affect the speed of the water waves, and thus, their wavelength.

## Procedure

• David McKay

Ivy league graduate, 4+ yrs of experience, vetted IB Examiner!

• ### Apparatus

• Lascells Ripple – Strobe Tank
• Stroboscope (included in the Lascells Ripple Tank)
• Thin wire able to generate plane water waves
• 1-millimeter Grid Graph Paper
• A predesigned metal strip
• Two metal plates (2.5 cm × 2.5 cm)
• Water Tank (10 cm × 10 cm × 2 cm)
• Water
• 500 - ml Graduated Cylinder (± 5 ml)
• Vernier Caliper (± 0.01 cm)

Figure 3 -

### Methodology

• Add the two metal plates in the water tank and place the predesigned metal strip as shown on the diagram and at a distance of 1.80 ± 0.01 cm from the start of the tank
• Fill the graduated cylinder with 115 ml of water and pour it into the water tank
• Insert the wire in the ripple motor and place it in such a way that all its flat part responsible for generating plane water waves touches the water in the tank
• Turn on the ripple motor and set it to such a frequency that the wavelength of the generated plane water waves is 1.0 ± 0.01 cm
• Make sure that the wavelength is 1.0 ± 0.01 cm by flashing the stroboscope light at the same frequency as the ripple motor and measuring the distance between the center of two wavefronts to be 1.0 ± 0.01 cm. Adjust the frequency of the ripple motor if this is not the case. Since it is possible that, depending on the two wavefronts considered, slightly different wavelength values are obtained, measure the mean wavelength, and make sure that it closely approximates to 1.0 ± 0.01 cm.
• Place each metal plate vertically and on one side of the predesigned metal strip. Move both plates to create a single slit aperture of length 0.4 cm. Measure the length of the aperture with the vernier caliper. Make sure that both sides that create the single slit aperture are symmetrical so that the shadow regions produced on both sides of the tank are closely similar.
• Flash the stroboscope light at the same frequency as that of the ripple motor. The ‘frozen’ wavefronts should be projected in the screen of the ripple tank. If necessary, stop any sunlight from coming into the room and turn off all artificial lights to improve how wavefronts are projected.
• Place graph paper on the screen. Plot specific coordinates where the first notorious disparity in wave amplitude and intensity seems to appear within each of the first seven diffracted wavefronts. Coordinate (0,0) will be considered to be the end of the first diffracted wavefront.
• Use mathematical software to draw a line of best fit for the plotted coordinates within each diffracted wavefront. Then, calculate the angle of diffraction by determining the slope of the line of best fit and applying the formulae seen above.
• Repeat steps 6-10 for single slit length values of 0.6 cm, 0.8 cm, 1.0 cm, 1.2 cm, 1.4 cm, 1.6 cm, 1.8 cm, and 2.0 cm
• David McKay

Ivy league graduate, 4+ yrs of experience, vetted IB Examiner!

• ## Raw Data

The following tables contain the specific coordinates plotted at the left and right-hand side end of the first seven diffracted wavefronts for each single slit length.

Figure 5 - Table On The Specific Coordinates Plotted At The Left - Hand End
Figure 6 - Table On The Specific Coordinates Plotted At The Right - Hand End

Coordinates were plotted for only the first seven diffracted wavefronts since the succeeding diffracted wavefronts were hardly visible as their amplitude substantially decreased while propagating through the tank. Thus, the accuracy of the measurements was increased by ignoring wavefronts where establishing the first clear disparity in wave intensity and amplitude within them was unclear.

Each coordinate represents the horizontal (x value) and vertical (y value) distance, in millimeters, between the determined end of a diffracted wavefront and the end of the first diffracted wavefront. The uncertainty of the x and y value of each coordinate will be considered to be ±1 mm so as to take into account both the systematic error of the - millimeter grid graph paper and the possible human error when plotting the specific coordinates.

The raw data obtained can be deemed to be reliable, as the expected patterns indicated in the background theory can be observed. For instance, it was predicted that, as the single slit length increases, the horizontal distance between the end of the first and the last diffracted wavefront would decrease since the angle of diffraction is expected to decrease. For this same reason, the vertical distance between the end of the first and the last diffracted wavefront was expected to increase as the angle of diffraction decreases. This overall pattern can be observed in the raw data. For example, as the single slit length increases, the y-value of the coordinate at the end of the seventh diffracted wavefront generally increases while the x-value gradually decreases. Moreover, it can be noted that, as the single slit length increases, the change in both the x and y-value of the seventh coordinate gradually decreases, which supports the hypothesis that the angle of diffraction is expected to change to a lesser extent for larger single slit lengths.

Nevertheless, an important observation about the data is the fact that the vertical distance between the end of the first and the last wavefront tends to be greater for right-hand side values than for left-hand side ones. This is even more notorious in the coordinates plotted for the 0.80 cm single slit length where there is a substantial disparity between the y-coordinate of the right-hand side end (32) and that of the left-hand side end (19). A possible explanation for this can be the human error when placing the two metal plates to create a 0.80 cm single slit aperture. For example, perhaps there was not a precise symmetry between the left and the right-hand side after placing the metal plates. Another possible explanation can be the qualitative observation that the plane water wavefronts generated by the thin wire connected to the ripple motor were not totally linear but slightly slanted to the right hand side. Furthermore, the left-hand side part of the diffracted wavefronts was slightly more noticeable than the right-hand side, which suggests that the intensity of the diffracted wavefronts was greater, to a certain extent, for their left-hand side. All this could have slightly affected the overall diffraction pattern in each side of the ripple tank, thus creating a slight asymmetry between the values of both sides.

## Processed Data

The following table shows the left and right-hand side processed results, as well as the average angle of diffraction, each with their respective absolute uncertainties.

Figure 7 - Table On The Left And Right - Hand Side Processed Results As Well As The Average Angle Of Diffraction Each With Their Respective Absolute Uncertainties.

From the table above, it can be noted that the angle of diffraction for the left and right-hand side, for a given single slit length, generally overlap after having considered their absolute uncertainty. This suggests that, since the left and right-hand side measurements obtained can be considered to be close in value, the overall results are reliable, as physical theory suggests a clear symmetry between both sides. Additionally, the fact that the percentage uncertainty obtained in the average angle of diffraction is generally less than 30% also indicates that the overall results are reliable. Nevertheless, for some single slit lengths, such as 0.80 cm or 1.00 cm, the values calculated for both the line of best fit slope and angle of diffraction are notoriously different between the left and right-hand side. This indicates that some human error was present in the experiment when intending to ensure a symmetry between both sides, in spite of the considerations taken to mitigate this type of error.

### Justification of absolute uncertainties

All uncertainties were rounded up so as to not ignore the implications of the initial uncertainty value encountered. Furthermore, it should be noted that the absolute uncertainty of the slope and of the angle of diffraction measurements was presented to the second decimal place. This is to better represent the range of the results obtained since, for instance, deciding to present uncertainties to only one significant figure might lead to a misleading interpretation of the results, especially for the slope encountered for small single slit values like 0.4 cm or 0.6 cm and the angle of diffraction obtained for large single slit lengths like 1.8 cm and 2.0 cm.

The uncertainty of the slope was calculated for each given measurement by finding both the least and the greatest slope possible that connect the corners of the first and last error boxes for a given set of coordinates. Then, the following formula was applied -

$$\Delta Slope=\frac{{\text{Maximum Slope - Minimum Slope}}}{2}$$

The absolute uncertainty of the left and right-hand side angle of diffraction was calculated by applying the following error propagation formula -

$$s_{\bar{z}}\approx\sqrt{\sum^m_{j=1}(\frac{∂z}{∂x_j}s_{\bar x_j})^2}$$

For this particular case, it leads to -

$$\Delta\theta=|\frac{\Delta Slope}{1+(Slope)^2}|$$

Finally, the uncertainty of the average angle of diffraction for a given single slit value was calculated by adding half the absolute uncertainty of the left hand-side angle of diffraction measurements with half the absolute uncertainty of the right- hand side measurements.

### Graphs

Figure 8 - The Variation Of The Average Angle Of Diffraction With The Single Slit Length.

The average angle of diffraction was graphed since, as shown in the background theory, left and right-hand side results are expected to be close in value. Thus, averaging the results obtained for both sides intends to graph values that are within those obtained for the left and right-hand side, which will ultimately improve the accuracy of the analysis. Moreover, it should be noted that the horizontal error bars in the graph are almost unnoticeable due to their low value (± 0.01 cm) compared to the x-axis measurements and graph scale. Furthermore, it is also noteworthy to mention that, since the wavelength in this investigation was kept constant at 1.00 cm, the graph also represents the relationship between the average angle of diffraction and the aperture-wavelength ratio, as dividing the single slit length by the wavelength would not affect the results obtained.

The relationship observed in the graph was best approximated by the exponential equation (of form y = aebx) y = 2.8333e−1.548x. The expected relationship expressed in the hypothesis, where there is a generally negative slope that gradually diminishes as the single slit length increases, is modelled by this equation, and it can also be observed in the graph. This shows that the results obtained are, generally, on par with the background theory. Moreover, the high value of the coefficient of determination (R2 = 0.9575) indicates that, for the results obtained, there is a strong relationship between the angle of diffraction and the length of the single slit. In this way, it can be suggested that the results obtained in this investigation can be representative, to some extent, about how the angle of diffraction decreases as the single slit length, as well as the aperture-wavelength ratio, increases for the range of values previously determined.

Nevertheless, it should be noted that the exponential equation encountered passes thoroughly to practically all the error bars of single slit length measurements greater than 1.0 cm, while for single slit lengths smaller than 1.0 cm, there appear to be some anomalies that lead the graph to not even touch the error bars of single slit measurements of 0.6 cm and 0.8 cm. This implies that the reliability and accurateness of the measurements for single slit lengths smaller than 1.0 cm might be appreciably less than that of single slit lengths greater than 1.0 cm, which suggests a greater impact of human error when carrying out the methodology for these smaller single slit lengths. Notwithstanding, since the exponential trend can also be observed for these single slit values, it can be noted that the reliability of the measurements was not drastically (but only slightly) hampered, as after all, the exponential equation y = 2.8333e−1.548x still leads to high coefficient of determination, and thus, to a substantially strong relationship for the processed data.

On the other hand, an important observation to be made is the fact that none of the angle of diffraction measurements calculated, in spite of obtaining results for a single slit length value that is less than half the wavelength, were considerably close in value to $$\frac{\pi}{2}$$ rad. This contradicts some academic literature, such as academic consultant Paul Andersen’s explanation on wave diffraction8 that predicts a pronounced diffraction pattern – where the angle of diffraction value is very close to $$\frac{\pi}2$$ rad – would be obtained for single slit length values close to or slightly less than the wavelength. Additionally, the scientific context generally supports the idea that a single slit length value that is less than half the wavelength is expected to clearly lead to easily visible wavefronts and an undoubted $$\frac{\pi}{2}$$ rad angle of diffraction, which is not observed in the results obtained.

By substituting $$y=\frac {\pi}{2}$$ rad in the graph’s exponential equation and solving for x, one can notice that it is predicted that a single slit length value slightly smaller than 0.4 cm (x specifically equals 0.38 cm) will generate a $$\frac {\pi}{2}$$ rad angle of diffraction. This suggests that for this experiment, clearly pronounced diffraction patterns, which the academic literature indicates to start happening when the single slit length is comparable to the wavelength, are expected to occur when the single slit length is less than two fifths the actual wavelength. This is an intriguing result since the scientific context does not state or support any direct relationship between the wavelength and a single slit aperture that is less than half the wavelength.

Figure 9 - The Relationship Between The Average Angle Of Diffraction And The Single Slit Length Including A Line Of Best Fit And Maximum And Minimum Slope Lines.

For this graph, the absolute uncertainty of ln (Average Angle of Diffraction) was calculated, for each measurement, by applying the formula -

$$\Delta In (\theta)= |\frac{\Delta\theta}{\theta}|$$

Moreover, it should be noted that the horizontal error bars, as in the graph of Average Angle of Diffraction against Single Slit Length, are negligible since the value of the absolute uncertainty (± 0.01 cm) is substantially low compared to the graph scale, as well as to the x-axis measurements.

As observed in the linearized graph, the line of minimum or maximum slope pass through most of the graph’s error bars. This suggests that, for most measurements, the presence of random or human errors did not have a significant impact on the accuracy of the experiment. Nevertheless, for single measurements of 0.6 cm and 0.8 cm, the error bars of the angle of diffraction lie slightly above the line of best fit or the maximum slope line. This indicates that, for these two measurements, there might have been the presence of human error that slightly increased the value of the angle of diffraction.

To analyze the impact systematic and random errors might have had in the experiment, the percentage uncertainty of the slope of the line of best will be calculated -

$$\Delta Slope =\frac{{\text{Maximum Slope - Minimum Slope}}}{2}=\frac{-1.221-(-1.7924)}{2}=0.2857\frac{rad}{cm}$$

$${\text {Percentage Uncertainty}}=\frac{\Delta Slope}{|Slope|}×100=18.49 {\%}$$

The 18.49% uncertainty of the line of best fit slope shows that the results obtained in this investigation are generally reliable, as sources of errors did not have a notorious impact in the results. Thus, it is suggested that, in spite of the presence of systematic, random, and human errors in the methodology, the processed data remained reliable, which led to a more accurate analysis, evaluation, and conclusion.

To strengthen this analysis, we can also calculate the uncertainty of parameter a in the exponential equation encountered (y = 2.8333e−1.548x) so as to compare it with the uncertainty of parameter b and thus further evaluate the impact of possible sources of error in the investigation -

y = 2.8333e-1.548x = ae-bx

In(y) = In(2.8333) - 1.548x = In(a) - bx

In(a) = In(y) + bx = u          (u - Substitution)

$$\Delta In (a)=\frac{{\text{Maximum possible Value of In(a) - Minimum possible value of In(a)}}}{2}= \frac{1.1419-0.7992}{2}={\text{0.17135 rad}}$$

$$∴ \Delta a=\Delta e^u=\sqrt{(\frac{d(a)}{du}×\Delta u)^2}=|e^u× \Delta u|=a×\Delta In (a) = (2.8333)(0.17135)={\text{0.4855 red}}$$

$${\text{Percentage uncertainty of a}}=\frac{\Delta a}{a}×100 = \frac{0.4855}{2.833}×100=17.14\%$$

Thus, it can be observed that parameter a has a considerably low percentage uncertainty that is very close in value to the percentage uncertainty of parameter b (18.49%). In this way, the results obtained in this investigation can be deemed to be generally reliable, and it can be indicated that the sources of error throughout the experiment, whether involving systematic, random, or human error, did not have a substantial impact in the reliability of the results.

• David McKay

Ivy league graduate, 4+ yrs of experience, vetted IB Examiner!

• ## Evaluation

### Analysis and Conclusion

Ultimately, this investigation allowed me to explore and address the research question “How does the length of a single slit aperture in a ripple tank affect the angle of diffraction generated after plane water waves pass through the single slit?”. As observed by the graph of average angle of diffraction vs. single slit length, the equation that best models the encountered relationship is an exponential relationship that resembles an inverse correlation, asthe slope of the graph is always negative and gradually decreases as the single slit length increases. Moreover, since the wavelength in this experiment was set to be 1 cm, the results also represent and model the relationship between the average angle of diffraction and the aperture- wavelength ratio. This provides greater insight about the nature of single slit diffraction since the relationship between the length of the slit and the wavelength is a pivotal one in determining the extent to which the diffraction pattern is pronounced.

Nevertheless, although the relationship encountered is on par with the background theory, the results obtained for single slit lengths smaller than the wavelength contradict physical theory and academic literature, as it is generally predicted that an angle of diffraction close to $$\frac{\pi}{2}$$ rad will be produced for single slit values comparable to or slightly greater than the wavelength. Moreover, a truly pronounced diffraction pattern was not even observed when recording data for the single slit length 0.4 cm, which is smaller than half the wavelength, and thus, further opposes to the academic literature and physical theory that indicate that an astoundingly clear pronounced diffraction pattern should be observed for this aperture- wavelength ratio. Nevertheless, it can be noted that, for this experiment, a single slit length slightly smaller than 0.4 cm is predicted to produce a $$\frac{\pi}{2}$$ rad angle of diffraction, as observed by substituting $$y=\frac{\pi}{2}$$ rad in the exponential equation that best models the angle of diffraction – single slit length relationship (2.8333e−1.548x) and solving for x. This is an even more intriguing result as the scientific context does not state or support any direct relationship between a single slit length smaller than two fifths the wavelength and the extent to which a single slit diffraction pattern is pronounced.

The results obtained in this investigation can be regarded as generally accurate and reliable. For instance, in spite of some outliers, the raw data obtained shows the patterns predicted by the background theory and the hypothesis (namely the decrease of the horizontal distance and the increase of the vertical distance between the coordinates plotted for the first and the last diffracted wavefront as the single slit length increased), which shows that the results are on par with other scientific works and studies on the topic. Moreover, this is also supported by the closeness in value between the left and the right- hand side angle of diffraction measurements in the processed data table, as it is also a result that is according to the background theory, and thus, indicates that any source of error when plotting the coordinates was not significant enough to hinder the reliability of the measurements. Furthermore, although human error might have played a greater impact on the measurements for single slit lengths smaller than 1.0 cm, as observed by the fact that the exponential equation in the graph of average angle of diffraction against single slit length was not even close to touch the error bars of single slit lengths 0.6 cm and 0.8 cm, the overall impact of the sources of errors in the results obtained was determined to be low, as noted by the strong exponential relationship encountered and the relatively low percentage uncertainty of the parameters a (17.14%) and b (18.49%) of the encountered exponential equation 2.8333e−1.548x.

Ultimately, the methodology utilized played a crucial role in further enhancing the reliability of the results, as it led to a much more realistic approach to measuring and recording data. For instance, the reliability of the experiment was enhanced by the fact that specific coordinates were plotted at the end of each diffracted wavefront, instead of simply drawing a diagonal line that seemingly connects all these points. Furthermore, utilizing formulae to calculate the left and right-hand side angle of diffraction considerably increased the accuracy of the investigation by eliminating possible human error when measuring this angle through other methods.

### Weaknesses and Improvements

One of the main weaknesses and source of errors in this investigation was the fact that the results obtained for the left and right-hand side signal a slight discrepancy and asymmetry between both sides. This could have been caused by the human error when placing the metal plates to generate a single slit aperture, as well as by the fact that the wavefronts were slightly slanted to the right. The impact this factor had on the investigation was considerable, as it hindered the accuracy of the results by generating a slight asymmetry and some discrepancies when comparing the left and right-hand side. These slight discrepancies between both sides underscore the presence of both systematic and human errors, which illustrate the impact of this factor on the accuracy of the data collected. A possible way to improve upon this factor can be crafting symmetrical predesigned metal strips for each single slit length, rather than moving two metal plates to generate all single slit apertures (thus reducing possible human error). Moreover, the plane water waves could be generated by an oscillating paddle, instead of by a thin wire, so as to reduce the systematic error by decreasing the extent to which wavefronts appear to be slanted.

In addition, another factor that decreased the accuracy of the results obtained was the fact that the wavefronts were barely observed at the end of the water tank, as their amplitude substantially decreases as they propagate through the tank. This led to plotting coordinates for only the first seven wavefronts and ignoring about one quarter the total diffracted wavefronts since it was unclear and ambiguous to plot specific coordinates at the end of those wavefronts. Thus, the data encountered is limited to only considering about three fourths of the generated diffraction pattern, which reduces the accuracy of the investigation. This factor can be improved by utilizing different ripple motors that are able to generate plane water waves with a much greater initial amplitude.

• David McKay

Ivy league graduate, 4+ yrs of experience, vetted IB Examiner!

• ### Extensions

A possible extension to this investigation would be researching whether the encountered relationship, as well as the specific single length value that generates an angle of diffraction very close in value to $$\frac{\pi}{2}$$ rad, holds for a much wider range of aperture-wavelength ratio. For instance, the single slit length value could range from being ten times smaller than the wavelength to ten times greater. Thus, it would be possible to analyze at which exact aperture-wavelength ratio the angle of diffraction would be equal to $$\frac{\pi}{2}$$ rad, as well as at which it equals 0 rad. Moreover, as more data would be collected, one would be able to determine a much more precise coefficient of determination and a much more precise equation that models the relationship between the angle of diffraction and the single slit length. Another possible extension to this experiment could be investigating whether the conclusions drawn apply if we keep the single slit length constant while changing the wavelength. Once again, a wide range of aperture-wavelength ratio would be ideal, as it would enable us to explore the relationship for a wider range of data and to determine when does the angle of diffraction approximates to both 0 and $$\frac{\pi}{2}$$ rad. Thus, it would lead to more substantiated results and conclusions. It would then be interesting to compare the results found for these conditions with those when the wavelength was kept constant while the single slit length varied, as thus, we would be able to explore whether different results are obtained for similar aperture-wavelength ratios.

## Bibliography

“Diffraction.” Libretexts. November 5, 2020. https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_Physics_(Boundless)/26%3A_Wave_Optics/26 .2%3A_Diffraction.

“Diffraction and Interference.” http://electron6.phys.utk.edu/phys250/modules/module%201/diffraction_and_interference.htm.

Kirchner, James. “Data Analysis Toolkit #5: Uncertainty Analysis and Propagation.” PDF File. http://seismo.berkeley.edu/~kirchner/eps_120/Toolkits/Toolkit_05.pdf.

“Identifying the Diffraction Angle of Light Waves.” Nagwa. https://www.nagwa.com/en/videos/642124274931/. https://torrie.edublogs.org/files/2014/09/what_to_do_with_max_and_min_lines-2014-1b1dyma.pdf.

• David McKay

Ivy league graduate, 4+ yrs of experience, vetted IB Examiner!