# Evaluating Standing Waves

1. For a particular tube there are six harmonic frequencies below 1000 Hz. Four of these are 300Hz, 600Hz, 750Hz, and 900Hz. You can see that one end of the tube is open, but you can not see the other end. Is it open or closed? Explain your answer.

2. In the tube described above, what two frequencies are missing?

3. When a musical note is played on a large church pipe organ, the sound originates from blowing puffs of air into a tube of a fixed length. Does this puff of air have to have only the particular frequency of the final note? If so, how do you think the organ creates the puff with this frequency? If it does not, why you you think you only hear a singe frequency?

4. The audible frequency range for normal hearing is from about 20Hz to 20 kHz. What are the wavelengths of sound waves at these frequencies?

5. A small loudspeaker driven by an audio oscillator and amplifier, adjustable in frequency. Nearby is a tube of cylindrical sheet-metal pipe 45.7 cm long, with a diameter of 2.3 cm and which is open at both ends. If the room temperature is 20 degrees Celsius, at what frequencies will resonance occur in the pipe when the frequency emitted by the speaker is varied from 1000 to 2000 Hz.

6. a) What is the fundamental frequency of the tube in the previous problem?

b) What would be the fundamental frequency if we doubled the diameter of the to 4.6cm?

c) What would be the fundamental frequency if we doubled the length of the tube to 91.4cm?

d) What would be the fundamental frequency if we decreased the velocity of sound in air to 320 m/sec?

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#### Solution Preview

Here are answers to your problems:

1) An organ pipe which is open at both ends can produce fundamental of frequency f and harmonics of frequencies 2f,3f, 4f etc. That is even as well as odd harmonics. An organ pipe which is open at one end and closed at the other can produce only odd harmonics. That is, f, 3f, 5f etc., if f is fundamental.The pipe in question is open at both ends and is capable of producing a fundamental frequency of 150 Hz and harmonics of 300Hz, 450Hz, 600Hz, 750Hz and 900Hz(below 1000Hz). Of these, ...

#### Solution Summary

This solution provides a detailed, step-wise response, including all of the required calculations.

Nodes of a Standing Wave (Sine)

The nodes of a standing wave are points at which the displacement of the wave is zero at all times. Nodes are important for matching boundary conditions, for example, that the point at which a string is tied to a support has zero displacement at all times (i.e., the point of attachment does not move).

Consider a standing wave, where y represents the transverse displacement of a string that extends along the x direction. Here is a common mathematical form for such a wave:

y(x,t) = A sin(k x) sin(omega t),

where A is the maximum transverse displacement of the string (the amplitude of the wave), which is assumed to be nonzero, k is the wave number, omega is the angular frequency of the wave, and t is time.

a). At time t=0, what is the displacement of the string y(x,0)?

Express your answer in terms of A, k, and other previously introduced quantities.

b). What is the displacement of the string as a function of x at time T/4, where T is the period of oscillation of the string?

Express the displacement in terms of A, x, k, and other constants; that is, evaluate omega frac{T}{4} and substitute it in the expression for y(x,t).

c). At which three points x_1, x_2, and x_3 closest to x=0 but with x>0 will the displacement of the string y(x,t) be zero for all times? These are the first three nodal points.

Express the first three nonzero nodal points in terms of the wavelength lambda. List them in increasing order, separated by commas. You should enter only the factors that multiply lambda. Do not enter lambda for each one.