Displacement and pressure in sound wave

[denian]

there is one formula given as follow

(max pressure ) = (speed of wave)X(density of medium)X(angular frequency of the wave ) X (maximum displacement)

in the book, there is no working shown how to arrive to this formula. anyone can show me the steps? thank you.


(sorry, i don't know how to insert the symbols)

 

[himanshu121]

consider

$$s=s_m\sin(kx-\omega t)$$

$$\Delta p=-B\frac{ds}{dx}$$

$$\Delta p(x,t) = Bks_m\sin(kx-\omega t)$$

$$B=\rho v^2$$

$$k=\frac{\omega}{v}$$

hence u get
$$\Delta p_m=\omega v \rho s_m$$

 

[M98Ranger]

The formula given is known as the acoustic pressure formula and it is derived from the basic principles of wave motion. To understand how this formula is derived, we need to first understand the concepts of displacement and pressure in sound waves.

Displacement refers to the distance that a particle in the medium moves from its equilibrium position when a sound wave passes through it. This displacement can be positive (when the particle moves away from its equilibrium position) or negative (when the particle moves towards its equilibrium position).

Pressure, on the other hand, refers to the force per unit area that the sound wave exerts on the medium as it propagates through it. When a sound wave passes through a medium, it compresses and rarefies the particles in that medium, causing changes in pressure.

Now, let's break down the formula step by step:

1. (Maximum pressure): This refers to the maximum pressure that the sound wave exerts on the medium. This pressure is directly proportional to the maximum displacement of the particles in the medium. In other words, the greater the maximum displacement, the greater the maximum pressure.

2. (Speed of wave): This refers to the speed at which the sound wave travels through the medium. This speed is determined by the properties of the medium, such as its density and elasticity. Generally, sound waves travel faster in denser and more elastic mediums.

3. (Density of medium): This refers to the density of the medium through which the sound wave is propagating. Density is defined as the mass per unit volume of a substance. The denser the medium, the more particles there are to compress and rarefy, resulting in a greater pressure.

4. (Angular frequency of the wave): This refers to the frequency at which the sound wave oscillates in a circular motion. It is related to the regular frequency of the wave (measured in Hertz) by the formula: angular frequency = 2π x frequency. The higher the frequency or angular frequency, the more cycles of compression and rarefaction occur in a given time, resulting in a greater pressure.

5. (Maximum displacement): As mentioned earlier, this refers to the maximum displacement of the particles in the medium as the sound wave passes through it. This displacement is directly proportional to the maximum pressure exerted by the sound wave.

By combining all these factors, we arrive at the acoustic pressure formula: (Max pressure) = (Speed of wave) x (Density of medium) x (Angular frequency