Question

A whistle is often used in dog-­‐‑training exercises and is an integral part of field-­‐‑marking competitions, where signals and commands are relayed to the dog via whistle. Suppose the whistle produces a sound wave with a frequency of 25,100 Hz (outside the range of human hearing) and the propagation speed of sound in air is 343 m/s.

1) What is the wavelength and wave number of this sound from the whistle?
2) Write the wave function for the longitudinal sound wave described above, assuming the amplitude of the sound wave is Smax = 1.57 x 10-­‐‑6 m and it is moving to the right (in the positive x direction).

Answer 1

The wavelength is
`\lambda = 0.01367 m`

The wave number is
`N = 73.18\ m^(-1)`

The wave function is
`y= 1.57 *10^(-6) sin 2 \pi ( 73.178 x -25100t)`

Step-by-step explanation:

From the question we are told that

The frequency of the sound wave is
`f = 25,100 Hz`

The speed of the wave is
`v = 343 m/s`

The wavelength of the wave is mathematically evaluated as

`\lambda = (v)/(f)`

substituting values

`\lambda = (343)/(25100)`

`\lambda = 0.01367 m`

The wave number is mathematically represented as

`N= (1)/(\lambda )`

substituting values

`N = (1)/( 0.01367 )`

`N = 73.18\ m^(-1)`

The general form of wave function is

`y= A sin (kx -wt)`

given that the amplitude is
`A = 1.57*10^(-6) \ m`

While
`w` which is the angular velocity is represented as
`w = 2 \pi f`

and k which is the angular wave number is mathematically represented as
`k = (2 \pi )/(\lambda )`

The wave function becomes

`y= A sin 2 \pi ((1)/(\lambda) x -ft)`

substituting values

`y= 1.57 *10^(-6) sin 2 \pi ( 73.178 x -25100t)`