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

Answer:

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

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

$f=\dfrac{v}{\lambda}$

Put the value into the formula

$f=\dfrac{343}{0.190}$

$f=1805.2\ Hz$

We need to calculate the angular frequency

Using formula of angular frequency

$\omega =2\pi f$

Put the value into the formula

$\omega=2\pi\times1805.2$

$\omega=11342.40\ rad/s$

We need to calculate the wave number

Using formula of wave number

$k = \dfrac{2\pi}{\lambda}$

Put the value into the formula

$k=\dfrac{2\pi}{0.190}$

$k=33.06$

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