Calculating Frequency, Energy & Power of Sound Waves

[Werg22]

The human tympanum is displaced by sound waves. This displacement is linked to the acting force by the equation:

F = 4x

where x is the displacement in meters.

A wave of frequency 1000 Hz provoques a displacement of 1.5 x 10^-5 m. Prove that the energy given to the tympanum is given in 0.25*10^-3 s. Also find the average power of the wave.

I don't how power is linked to frequency and energy in therms of waves...

 

[andrevdh]

The intensity of a wave is defined as
$$I=\frac{P}{S}$$
where $P$ is its power (the amount of energy it transports per second) through a (perpendicular) cross sectional area $S$.

The intensity is given by
$$I=2\pi^2v\rho f^2A^2$$
which includes the propagation speed of the wave, $v$, while it is traveling throught a medium of density $\rho$, creating a disturbance with a frequency $f$ and amplitude $A$.

 

[kyle8921]

I can explain the relationship between frequency, energy, and power in sound waves. Frequency is the number of oscillations or cycles of a sound wave per second, measured in Hertz (Hz). Energy is the ability to do work and is measured in Joules (J). Power is the rate at which energy is transferred or the amount of work done per unit time, measured in Watts (W).

In this scenario, the human tympanum is displaced by a sound wave with a frequency of 1000 Hz, which means that the sound wave is oscillating at a rate of 1000 cycles per second. The displacement of the tympanum, x, is 1.5 x 10^-5 m.

To calculate the energy given to the tympanum, we can use the equation:

E = 0.5 * m * ω^2 * x^2

where E is the energy, m is the mass of the object (in this case, the tympanum), ω is the angular frequency (2πf) and x is the displacement.

Substituting the values given in the problem, we get:

E = 0.5 * m * (2πf)^2 * x^2

= 0.5 * m * (2π*1000)^2 * (1.5 x 10^-5)^2

= 0.5 * m * 1.2566 * 10^-1 * 2.25 * 10^-10

= 1.125 * 10^-12 * m

To find the energy given to the tympanum in 0.25 * 10^-3 s, we need to multiply the energy by the time, which gives us:

E = 1.125 * 10^-12 * m * 0.25 * 10^-3

= 2.8125 * 10^-16 * m

Therefore, the energy given to the tympanum is 2.8125 * 10^-16 J.

Now, to find the average power of the wave, we can use the equation:

P = E / t

where P is the power and t is the time.

Substituting the values, we get:

P = 2.8125 * 10^-16 J / (0.25 * 10^-3 s)

= 2.