Question

Create a mathematical model for the pressure variation as a function of position and time for a sound wave, given that the wavelength of the wave is λ = 0.190 m and the maximum pressure variation is ΔPmax = 0.270 N/m2. Assume the sound wave is sinusoidal. (Assume the speed of sound is 343 m/s. Use the following as necessary: x and t. Assume ΔP is in Pa and x and t are in m and s, respectively. Do not include units in your answer.)

Answer 1

The equation of position and time for a sound wave is
`\Delta p=0.270(33.06 x-11342.40 t)`.

Step-by-step explanation:

Given that,

Wavelength = 0.190 m

Maximum pressure
`\Delta P_(max)= 0.270 N/m^2`

We know that,

The function of position and time for a sound wave,

`\Delta p=\Delta p_(max)(kx-\omega t)`....(I)

We need to calculate the frequency

Using formula of frequency

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

Put the value into the formula

`f=(343)/(0.190)`

`f=1805.2\ Hz`

We need to calculate the angular frequency

Using formula of angular frequency

`\omega =2\pi f`

Put the value into the formula

`\omega=2\pi*1805.2`

`\omega=11342.40\ rad/s`

We need to calculate the wave number

Using formula of wave number

`k = (2\pi)/(\lambda)`

Put the value into the formula

`k=(2\pi)/(0.190)`

`k=33.06`

Now, put the value of k and ω in the equation (I)

`\Delta p=0.270(33.06 x-11342.40 t)`

Hence, The equation of position and time for a sound wave is
`\Delta p=0.270(33.06 x-11342.40 t)`.