Question

Show that the sound wave intensity \(I\) can be written in terms of the frequency \(f\) and displacement amplitude \(s_{\mathrm{m}}\) in the form $$ I=2 \pi^{2} \rho v f^{2} s_{\mathrm{m}}^{2} $$

Short answer

Substituting \(p = 2\pi f v \rho s_{\text{m}}\) into \(I = \frac{p^2}{2\rho v}\) yields the equation \(I = 2 \pi^{2} \rho v f^{2} s_{\mathrm{m}}^{2}\).

Step-by-step solution

Write down the formula for sound wave pressure

We know the formula for the pressure caused by a simple harmonic motion in a medium is given by \(p = s_{\text{m}}\omega v \rho\), where \(\omega\) is the angular frequency, \(v\) is the velocity of sound, \(\rho\) is the density of the medium and \(s_{\text{m}}\) is the displacement amplitude.

Express \(\omega\) in terms of \(f\)

We know that the angular frequency \(\omega\) is related to the frequency \(f\) by \(\omega = 2\pi f\). Substituting this to the equation from step 1 gives us \(p = 2\pi f v \rho s_{\text{m}}\).

Use the formula for Intensity of Sound Wave.

The sound wave intensity \(I\) is given by \(I = \frac{p^2}{2\rho v}\). Substituting the expression for \(p\) obtained from step 2 into this equation gives us \(I=\frac{(2\pi f v \rho s_{\text{m}})^{2}}{2\rho v}\).

Simplify Equation

We simplify the equation as follows: \(I=\frac{(2\pi f v \rho s_{\text{m}})^{2}}{2\rho v} = 2 \pi^{2} \rho v f^{2} s_{\mathrm{m}}^{2}\).

Key concepts

Frequency

The term frequency has a critical role in the realm of sound waves. Frequency, denoted as f, represents how often the particles of a medium vibrate when a wave passes through it. It is measured in hertz (Hz), where one hertz corresponds to one vibration per second. Higher frequencies are associated with higher-pitched sounds, while lower frequencies correspond to lower-pitched sounds. The frequency not only determines the pitch of the sound but also directly influences the intensity of the sound wave, as seen in our exercise where the sound wave intensity increases with the square of the frequency. This relationship has profound applications in various fields, ranging from music to medical imaging.

Displacement Amplitude

The term displacement amplitude, often symbolized by s_m, is a measure of how far the particles in the medium move from their rest position when a sound wave travels through it. Imagine it as the height of the wave as it oscillates. The greater this amplitude, the more energy the wave transports, which translates to a louder sound perceived by the human ear. In our exercise, the relationship between intensity and displacement amplitude is quadratic, meaning that sound intensity increases dramatically with an increase in displacement amplitude. This concept is key to understanding why whispers are faint while shouts are loud.

Angular Frequency

Angular frequency, usually denoted by ω, is a measure that expresses how quickly a sound wave oscillates in radians per second. Related directly to the more common frequency f (measured in cycles per second or hertz), it provides insight into the wave's oscillatory motion with the relationship ω = 2πf. This formula connects linear and angular motion, enabling us to link the displacement amplitude and frequency to derive an expression for the sound wave's pressure, as demonstrated in the steps for calculating intensity. Understanding the angular frequency is crucial for deeper insights into wave behaviors and their mathematical representations.

Pressure in Simple Harmonic Motion

In the context of sound waves, pressure refers to the alternating high and low-pressure regions created by the wave as it propagates through a medium. Sound waves arise from these pressure variations, producing simple harmonic motions of particles in the wave's path. This motion is characterized by its sinusoidal oscillation pattern. Our original problem illustrates how this rhythmic pressure motion is mathematically tied to the displacement amplitude and the wave's angular frequency. Moreover, understanding the relationship between pressure and sound wave properties, such as amplitude and frequency, is essential for applications such as acoustics engineering and audio technology.