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.

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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.

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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.

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: &ldquo;the bending of a wave around the edges of an opening or an obstacle.&rdquo; 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.

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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&nbsp;\(\frac{\pi}{2}\) 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&nbsp;wavefronts will reach places that seemed unreachable at a first glance.

Labelled Diagram Of A Pronounced Single Slit Diffraction Pattern.svg

The extent to which a single slit diffraction pattern is pronounced greatly depends on the relationship between the wavelength (&lambda;) and the length of the single slit. Physical theory explains that a wave&rsquo;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&nbsp;\(\frac{\pi}{2}\) 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.

A Comparison Between The Diffraction Patterns Generated By a Small Single Slit And A Large Single Slit.svg

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.

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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 &lsquo;shadow region&rsquo;). 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.

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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 &lsquo;the end&rsquo; of each diffracted wavefront.

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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&rsquo;s frequency, the wave&rsquo;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.

In this experiment, the range of the graph of angle of diffraction (&theta;) against single slit length would be 0 rad&nbsp;\(≤ θ ≤ \frac{π} 2\) rad, as &theta; cannot exceed an angle of&nbsp;\(\frac{\pi}{2}\) rad or go below a 0 rad angle. Moreover, &theta; is expected to be&nbsp;\(\frac{\pi}{2}\) 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.

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 &plusmn; 0.01 cm for this experiment).

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 -

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\(θ = \frac{π}2\)&nbsp;&minus; arctan(slope)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;if slope &gt; 0

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\(\theta=arctan(slope)+\frac{\pi}2\)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;if slope &lt; 0

The wavelength of the water waves (1.00 &plusmn; 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.

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The water depth will be controlled by pouring a specific amount of water (115 &plusmn; 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.

<ul>
	<li>Lascells Ripple &ndash; Strobe Tank</li>
	<li>Stroboscope (included in the Lascells Ripple Tank)</li>
	<li>Thin wire able to generate plane water waves</li>
	<li>1-millimeter Grid Graph Paper</li>
	<li>A predesigned metal strip</li>
	<li>Two metal plates (2.5 cm &times; 2.5 cm)</li>
	<li>Water Tank (10 cm &times; 10 cm &times; 2 cm)</li>
	<li>Water</li>
	<li>500 - ml Graduated Cylinder (&plusmn; 5 ml)</li>
	<li>Vernier Caliper (&plusmn; 0.01 cm)</li>
</ul>

Figure 3.svg

<ul>
	<li>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 &plusmn; 0.01 cm from the start of the tank</li>
	<li>Fill the graduated cylinder with 115 ml of water and pour it into the water tank</li>
	<li>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</li>
	<li>Turn on the ripple motor and set it to such a frequency that the wavelength of the generated plane water waves is 1.0 &plusmn; 0.01 cm</li>
	<li>Make sure that the wavelength is 1.0 &plusmn; 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 &plusmn; 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 &plusmn; 0.01 cm.</li>
	<li>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.</li>
	<li>Flash the stroboscope light at the same frequency as that of the ripple motor. The &lsquo;frozen&rsquo; 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.</li>
	<li>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.</li>
	<li>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.</li>
	<li>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</li>
</ul>

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.

Table On The Specific Coordinates Plotted At The Left - Hand End.svg

Table On The Specific Coordinates Plotted At The Right - Hand End.svg

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.

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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 &plusmn;1 mm so as to take into account both the systematic error of the 1&nbsp;- millimeter grid graph paper and the possible human error when plotting the specific coordinates.

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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.

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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.

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.

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

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.

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.

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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 -

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

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

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

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For this particular case, it leads to -

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

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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.

the variation of the average angle of diffraction with the single slit length..svg

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?