17 Sound
17.4 Normal Modes of a Standing Sound Wave
Learning Objectives
By the end of this section, you will be able to:
- Explain the mechanism behind sound-reducing headphones
- Describe resonance in a tube closed at one end and open at the other end
- Describe resonance in a tube open at both ends
Interference is the hallmark of waves, all of which exhibit constructive and destructive interference exactly analogous to that seen for water waves. In fact, one way to prove something “is a wave” is to observe interference effects. Since sound is a wave, we expect it to exhibit interference.
Interference of Sound Waves
In Waves, we discussed the interference of wave functions that differ only in a phase shift. We found that the wave function resulting from the superposition of [latex]{y}_{1}(x,t)=A\,\text{sin}(kx-\omega t+\varphi )[/latex] and [latex]{y}_{2}(x,t)=A\,\text{sin}(kx-\omega t)[/latex] is
One way for two identical waves that are initially in phase to become out of phase with one another is to have the waves travel different distances; that is, they have different path lengths. Sound waves provide an excellent example of a phase shift due to a path difference. As we have discussed, sound waves can basically be modeled as longitudinal waves, where the molecules of the medium oscillate around an equilibrium position, or as pressure waves.
When the waves leave the speakers, they move out as spherical waves (Figure). The waves interfere; constructive inference is produced by the combination of two crests or two troughs, as shown. Destructive interference is produced by the combination of a trough and a crest.
The phase difference at each point is due to the different path lengths traveled by each wave. When the difference in the path lengths is an integer multiple of a wavelength,
the waves are in phase and there is constructive interference. When the difference in path lengths is an odd multiple of a half wavelength,
the waves are [latex]180^\circ(\pi \,\text{rad})[/latex] out of phase and the result is destructive interference. These points can be located with a sound-level intensity meter.
Example
Interference of Sound Waves
Two speakers are separated by 5.00 m and are being driven by a signal generator at an unknown frequency. A student with a sound-level meter walks out 6.00 m and down 2.00 m, and finds the first minimum intensity, as shown below. What is the frequency supplied by the signal generator? Assume the wave speed of sound is [latex]v=343.00\,\text{m/s}\text{.}[/latex]
Strategy
The wave velocity is equal to [latex]v=\frac{\lambda }{T}=\lambda f.[/latex] The frequency is then [latex]f=\frac{v}{\lambda }.[/latex] A minimum intensity indicates destructive interference and the first such point occurs where there is path difference of [latex]\Delta r=\lambda \text{/}2,[/latex] which can be found from the geometry.
Solution
- Find the path length to the minimum point from each speaker.
[latex]{r}_{1}=\sqrt{{(6.00\,\text{m})}^{2}+{(2