Standing Waves

Sound tube (longitudinal waves) is on the left; white bungee cord and harmonic attached to harmonic oscillator is on the right.

Objective: To observe both transverse and longitudinal standing waves. To find the nodes and antinodes of a standing wave. To calculate the wavelength, frequency and velocity of a standing wave. To investigate how the velocity of a transverse wave on a string depends on mass density and tension.

Apparatus: Black sound tubes, long white microphones (plugged into speaker). Pasco Harmonic Oscillator, ring stand, thin white bungee cord, mass hanger, pulley, assorted masses, meter stick, weighing scale.

Theory:

Traveling waves have the form f(x-vt) or f(x+vt) where f stands for some function such as a sinusoid, e.g.

> sin(x-vt) or sin(x+vt).

The (x-vt) version represents a wave (state of affairs) traveling to the right (+ x direction is to the right), and the (x+vt) version represents a wave traveling to the left since (x-vt) stays the same when x increases while t does, and vise versa for (x+vt).

More generally, a sinusoidal wave could be written as

y = A sin(kx-wt)

where A is the amplitude, k is the “wave number” (# radians per meter at fixed time t) and w is the angular frequency (# radians per second at fixed position x). Equivalently, in terms of the temporal periodicity f (frequency in cycles/s or hertz (Hz) and spatial periodicity l (wavelength in meters)

y = A sin(2p x/l – 2p f t), and

f l = v (wave propagation velocity).

For a string under tension, v _string = sqrt( T/ r ) (T = tension, r = string mass/meter))

In an ideal gas, v sound is proportional to sqrt(T/m) (T = temperature K).

We will assume 20⁰ C, for which v air = 343 m/s.

Standing waves are a phenomenon of interference between two traveling waves of the same wavelength moving in opposite directions. Typically, a traveling wave is generated at one end of a linear system, and the oppositely moving traveling wave is obtained by reflection at the other end, only to be reversed again by reflection when it reaches the sending end. The standing wave pattern depends on the relative boundary conditions at the two ends: similar, producing integer number of half waves) or different, (producing odd-integer number of quarter waves, where the wavelength is the wave velocity divided by the frequency of sinusoidal excitation:

l = v/f .

For a string under tension, similar boundary conditions would be fixed-fixed (our experimental case) or free-free; different boundary conditions might be fixed-free or free-fixed. Waves in a string are transverse (involve particle motion perpendicular to direction of traveling wave propagation) might be either vertical or horizontal (two polarization directions). Waves in a fluid are longitudinal (individual particle motion in direction of traveling wave propagation.)

For a tube, similar boundary conditions would be open-open (o-o) or closed-closed (c-c); different boundary conditions would be open-closed (o-c) or closed-open (c-o).

For similar boundary conditions, possible standing wave patterns satisfy

L = n l / 2 (half waves, n = 1,2,3,4,------) or l sim = 2L/n

where L is the length of the bounded system, and

for different boundary conditions

L = (2n+1) l/4 (odd # of quarter waves, n = 1,2,3,4,--- or l dif = 4L/(2n+1) .

Procedure

Transverse Waves

Here you will use a mechanical oscillator to produce transverse waves on a thin white bungee cord. Note that the point where the cord touches the pulley is an antinode (minimum amplitude) and the point where the cords touches the harmonic oscillator is, to an approximation, also an antinode. Measure the distance between these two points and write it in the hand-in sheet.

1. Examine the apparatus - one end of the horizontal white bungee cord is attached to the mass hanger (over pulley); the other end is secured to the ring stand, and in between

(but close to the ring) is the harmonic oscillator facing up, ready to vibrate the string. Make sure there is enough mass on the mass hanger such that the string is taut enough to produce a wave if you pluck it. Also ensure that the mass is not too much that ring stand does not move and the mass hanger hangs too low (touches the ground). Note down the total mass hanger + added mass used in your hand-in sheet.

2. Open up "play.exe", which is function generator software in the same folder as this write-up. Press START; you should hear a sine wave sound immediately. The sound is coming from the harmonic oscillator, which is also vibrating the string. Lower the frequency using the sliding bar in the software, to something below 100 Hz. Feel the vibration on the string; if you are trouble doing so, increase the amplitude of vibration by turning up the volume knob on your PC speakers.

3. Decrease the frequency to near zero, and increase the frequency slowly (click on slider arrow or space to the left or right of slider) until you get a standing wave. Remember that a standing wave appears stationary to the naked eye - although there is obvious motion in the wave, the profile or shape of the wave does not change. Stop increasing the frequency when you see a clear standing wave. This is the fundamental frequency, or lowest frequency of natural vibration of the wave (also called the first harmonic). What fraction of a wavelength does this correspond to (where a whole wavelength looks like one complete sine cycle)? Write it in your hand-in sheet.

4. Increase the frequency further to find more frequencies (2nd harmonicsand above) at which standing wave develop. Note down these frequencies, and fractions of wavelenghts, in your hand-in sheet. Keep increasing the frequency and taking such data until you can no longer see any standing waves.

5. Repeat this for a diferent hanging weight value, perhaps double or half your original value. Record again in your hand-in sheet.

6. For your data analysis, you will need to know the linear mass density of the bungee cord. There are samples of short bungee cord next to the printer for this purpose. Write down the calculated linear mass density of the bungee cord in your hand-in sheet.

7. Write down your conclusion in the hand-in sheet about what frequencies/wavelengths are allowed on the bungee cord, at a particular tension. Also include in your conclusion a calculation of theoretical wave speed (from mass density and tension) and a calculation of experimental wave speed (from wavelength and frequency).

8. If your experimental results do not agree well with theory, consider that the bungee cord's linear density may change from what you calculated with the sample. Design an experiment that will lead to a better calculation of the linear mass density, re-do the experimetnal calculation and see how your results have changed. Record all your results in the hand-in sheet.

Longitudinal Waves

Here you will use your PC speaker to pump sound waves into the open end of a long black sound tube. This time the waves will not be visible to your eyes, but you will be able to detect the longitudinal pressure nodes and antinodes using a long microphone that you can slide inside the tube along its entire length. The tube can be open or closed at the microphone end, by the insertion or removal of the end cap. Record the length of the tube in your hand-in sheet.

The mike responds to pressure, not to longitudinal displacement, so the closed end should involve a maximum (antinode, A) and you should expect an odd number (2n+1) of quarter waves in the tube length for the resonant frequencies.

1. Make sure the end cap is in place. Open FFTScope, which is a template file in the same folder as this write-up. This software will enable you to generate standing waves in the tube and simultaneously monitor these waves using the microphone. In the application, go to Function Generator and select Sine. You should immediately hear the sine tone. The frequency can be varied by hitting Page Up/Page Down keys (10 Hz increments) or the Up/Down arrows keys (0.1 Hz increments). Note that FFTScope displays amplitude (y-axis) vs. frequency (x-axis).

2. Press the Go/Stop button in FFTScope to get the FFT spectrum analyzer started. You can click-drag the cursor on a lower range of frequencies to zoom. Insert the mike part way into the tube,. Press the Autoscale button (button with the four tiny red arrows), and vary the frequency up and down until you get a maximum FFT peak height in the graph. Record this frequency in your hand-in sheet.

3. Predict the harmonics associated with this resonant (fundamental) frequency from f ₁ l ₁ = f ₂ l ₂, etc.: (f ₂ = 3 f ₁, f ₃ = 5 f ₁ , etc. - odd integer multiples). Remember that the harmonics are given by the equation L = (2n+1) l/4 (odd # of quarter waves, n = 1,2,3,4,---) What harmonic frequencies do you actually observe in FFTScope? Write in hand-in sheet.

4. Tune to each harmonic (including the first harmonic, or fundamental), move the mike along the length of the tube and observe the variation in height of the drive frequency peak with FFTscope . What can you say about the distances between the nodes as you raise n (harmonic number)? Write your answer in the hand-in sheet.

Note on harmonic choice: It is easier to see the A's and N's for higher harmonics than for the fundamental or second. You might want to record #'s 3, 4, 5, 6 or try 4,5,7,8. It might be interesting to see how high you can go in harmonic number. When you start each harmonic, adjust the drive speaker for max amplitude. Keep the sound level as low as you can work well with.

5. Since nodes and antinodes are very difficult to pinpoint at low n, tune to the frequency of a harmonic of high n. Measure the number of nodes (or antinodes) in the tube and estimate the wavelength from your result. From the wavelength, calculate the frequency - does it indeed match the frequency you used? Show your calculation in the hand-in sheet.

6. Remove the end cap and repeat steps 3 & 4. Here, you should expect the ends to be nodes (N) of pressure, and half waves to fit the tube length L. The harmonics are now give my the equation L = n l / 2 (half waves, n = 1,2,3,4,----) What harmonic frequencies do you actually observe in FFTScope? Record all data in the hand-in sheet