Chapter 15, Problem 47P

(a)

To determine

To calculate:

Calculate the pressure amplitude of the wave.

Answer to Problem 47P

The pressure amplitude $\left({\text{p}}_{\text{0}}\right)$ of the wave is $\text{0}\text{.75}{\text{P}}_{\text{a}}$ .

Explanation of Solution

Given:

Pressure variation is given by,

  $\text{p}\left(\text{x,t}\right)\text{=0}\text{.75}\text{cos}\left[\frac{\text{π}}{\text{2}}\left(\text{x-343}\text{t}\right)\right]$

Formula used:

The pressure amplitude can be calculated by using:

  $\text{p}\left(\text{x,t}\right)\text{=0}\text{.75}\text{cos}\left[\frac{\text{π}}{\text{2}}\left(\text{x-343}\text{t}\right)\right]$

Where,

  $\text{k=}$ Wave number

  ${\text{p}}_{\text{0}}\text{=}$ Pressure amplitude,

  $\text{v=}$ Wave speed

The pressure variation is given as, $\text{p}\left(\text{x,t}\right)\text{=0}\text{.75}\text{cos}\left[\frac{\text{π}}{\text{2}}\left(\text{x-343}\text{t}\right)\right]$

Where,

  $\text{k=}$ Wave number

  ${\text{p}}_{\text{0}}\text{=}$ Pressure amplitude,

  $\text{v=}$ Wave speed

From the inspection of the equation which is given,

  $\text{p}\left(\text{x,t}\right)\text{=0}\text{.75}\text{cos}\left[\frac{\text{π}}{\text{2}}\left(\text{x-343}\text{t}\right)\right]$

The value of the ${\text{p}}_{\text{0}}\text{=0}\text{.75}{\text{P}}_{\text{a}}$

Conclusion:

Thus, the pressure amplitude $\left({\text{p}}_{\text{0}}\right)$ of the wave is $\text{0}\text{.75}{\text{P}}_{\text{a}}$ .

(b)

To determine

To calculate:

Calculate the wavelength of the wave.

Answer to Problem 47P

The wavelength $\left(\text{λ}\right)$ of the wave is $\text{4}\text{.00}\text{m}$ .

Explanation of Solution

Given:

Pressure variation is given by,

  $\text{p}\left(\text{x,t}\right)\text{=0}\text{.75}\text{cos}\left[\frac{\text{π}}{\text{2}}\left(\text{x-343}\text{t}\right)\right]$ .

Formula used:

The wavelength can be calculated by using:

  $\text{p}\left(\text{x,t}\right)\text{=0}\text{.75}\text{cos}\left[\frac{\text{π}}{\text{2}}\left(\text{x-343}\text{t}\right)\right]$

Where,

  $\text{k=}$ Wave number

  ${\text{p}}_{\text{0}}\text{=}$ Pressure amplitude

  $\text{v=}$ Wave speed

The pressure variation is given as,

  $\text{p}\left(\text{x,t}\right)\text{=0}\text{.75}\text{cos}\left[\frac{\text{π}}{\text{2}}\left(\text{x-343}\text{t}\right)\right]$

Where,

  $\text{k=}$ Wave number

  ${\text{p}}_{\text{0}}\text{=}$ Pressure amplitude

  $\text{v=}$ Wave speed

As the value of the $\text{k}$ ,

  $\text{k=}\frac{\text{2π}}{\text{λ}}\text{=}\frac{\text{π}}{\text{2}}$

Thus, $\text{λ=4}\text{.00}\text{m}$

Conclusion:

Thus, the wavelength $\left(\text{λ}\right)$ of the wave is $\text{4}\text{.00}\text{m}$ .

(c)

To determine

To calculate:

Calculate the frequency of the wave.

Answer to Problem 47P

The frequency $\left(\text{f}\right)$ of the wave is $\text{85}\text{.8}\text{Hz}$ .

Explanation of Solution

Given:

Pressure variation is given by,

  $\text{p}\left(\text{x,t}\right)\text{=0}\text{.75}\text{cos}\left[\frac{\text{π}}{\text{2}}\left(\text{x-343}\text{t}\right)\right]$ .

Formula used:

Frequency,

  $\text{f=}\frac{\text{kv}}{\text{2π}}$

Where,

  $\text{f=}$ Frequency of the wave

  $\text{v=}$ Wave speed

  $\text{k=}$ Wave number

The pressure variation is given as,

  $\text{p}\left(\text{x,t}\right)\text{=0}\text{.75}\text{cos}\left[\frac{\text{π}}{\text{2}}\left(\text{x-343}\text{t}\right)\right]$

Where,

  $\text{k=}$ Wave number

  ${\text{p}}_{\text{0}}\text{=}$ Pressure amplitude

  $\text{v=}$ Wave speed

Solve

  $\begin{array}{l}\text{v=}\frac{\text{ω}}{\text{k}}\\ \text{=}\frac{\text{2πf}}{\text{k}}\end{array}$

To get frequency,

  $\text{f=}\frac{\text{kv}}{\text{2π}}$

Substitute the numerical values in the above equation,

  $\begin{array}{l}\text{f=}\frac{\text{kv}}{\text{2π}}\\ \text{f=}\frac{\frac{\text{π}}{\text{2}}\left(\text{343}\text{m/s}\right)}{\text{2π}}\\ \text{f=85}\text{.8}\text{Hz}\end{array}$

Conclusion:

Thus, the frequency $\left(\text{f}\right)$ of the wave is $\text{85}\text{.8}\text{Hz}$ .

(d)

To determine

To calculate:

Calculate the speed of the wave.

Answer to Problem 47P

The speed $\left(\text{v}\right)$ of the wave is $\text{343}\text{m/s}$ .

Explanation of Solution

Given:

Pressure variation is given by,

  $\text{p}\left(\text{x,t}\right)\text{=0}\text{.75}\text{cos}\left[\frac{\text{π}}{\text{2}}\left(\text{x-343}\text{t}\right)\right]$

Formula used:

The pressure variation is given as,

  $\text{p}\left(\text{x,t}\right)\text{=0}\text{.75}\text{cos}\left[\frac{\text{π}}{\text{2}}\left(\text{x-343}\text{t}\right)\right]$

Where,

  $\text{k=}$ Wave number

  ${\text{p}}_{\text{0}}\text{=}$ Pressure amplitude

  $\text{v=}$ Wave speed

The pressure variation is given as,

  $\text{p}\left(\text{x,t}\right)\text{=0}\text{.75}\text{cos}\left[\frac{\text{π}}{\text{2}}\left(\text{x-343}\text{t}\right)\right]$

From the inspection of the equation which is given,

  $\text{p}\left(\text{x,t}\right)\text{=0}\text{.75}\text{cos}\left[\frac{\text{π}}{\text{2}}\left(\text{x-343}\text{t}\right)\right]$

The value of $\text{v=343}\text{m/s}$ .

Conclusion:

Thus, the speed $\left(\text{v}\right)$ of wave is $\text{343}\text{m/s}$ .s