Problem 73 - University Physics with Modern Physics

Question

A small sphere of radius \(R\) is arranged to pulsate so that its radius varies in simple harmonic motion between a minimum of \(R-\Delta R\) and a maximum of \(R+\Delta R\) with frequency \(f .\) This produces sound waves in the surrounding air of density \(\rho\) and bulk modulus \(B\) (a) Find the intensity of sound waves at the surface of the sphere. (The amplitude of oscillation of the sphere is the same as that of the air at the surface of the sphere.) (b) Find the total acoustic powcr radiated by the sphcrc. (c) At a distance \(d \gg R\) from the center of the sphere, find the amplitude, pressure amplitude, and intensity of the sound wave.

Step-by-Step Solution

Answer

At the sphere's surface, the intensity is given by \(\frac{1}{2} \rho A^2 \omega^2 \sqrt{\frac{B}{\rho}}\). The total acoustic power is \(4 \pi R^2 I\). At distance \(d\), amplitude is \(\frac{A R^2}{d}\), and intensity is \(\frac{P}{4 \pi d^2}\).

Step 1: Understanding Simple Harmonic Motion

The sphere undergoes simple harmonic motion with radius variations between minimum, \(R - \Delta R\), and maximum, \(R + \Delta R\). The amplitude \(A\) of this motion is \(\Delta R\). The angular frequency \(\omega\) is given by \(\omega = 2\pi f\), where \(f\) is the frequency. Thus, the radial motion can be described by \(R(t) = R + A \cos(\omega t)\).

Step 2: Calculate Surface Velocity

The velocity of the surface of the sphere is given by the derivative of the radius with respect to time: \(v(t) = \frac{d}{dt}(A \cos(\omega t)) = -A \omega \sin(\omega t)\). The maximum surface velocity \(v_{max}\) is \(A \omega\).

Step 3: Calculate Acoustic Intensity at Surface

The acoustic intensity \(I\) is given by \(I = \frac{1}{2} \rho v_{max}^2 c\), where \(c\) is the speed of sound in air, calculated as \(c = \sqrt{\frac{B}{\rho}}\). Substitute \(v_{max} = A \omega\) and \(c = \sqrt{\frac{B}{\rho}}\) into the formula to get the intensity at the surface: \[ I = \frac{1}{2} \rho (A \omega)^2 \sqrt{\frac{B}{\rho}} = \frac{1}{2} \rho A^2 \omega^2 \sqrt{\frac{B}{\rho}}. \]

Step 4: Calculate Total Acoustic Power

The total power radiated \(P\) by the sphere is given by \(P = 4 \pi R^2 I\), where \(I\) is the intensity at the sphere's surface. Substitute the expression for \(I\) from Step 3 to find \(P\): \[ P = 4 \pi R^2 \left(\frac{1}{2} \rho A^2 \omega^2 \sqrt{\frac{B}{\rho}}\right). \]

Step 5: Find Amplitude and Intensity at Distance d

At a distance \(d \gg R\), the amplitude of the sound wave \(a(d)\) decreases, and the intensity \(I(d)\) changes. The amplitude \(a(d)\) is proportional to \(\frac{A R^2}{d}\), and the intensity is inversely proportional to the square of the distance, \( I(d) = \frac{P}{4 \pi d^2}\).

Step 6: Calculate Pressure Amplitude at Distance d

The pressure amplitude \(p_0(d)\) at distance \(d\) can be calculated using the relation: \[ p_0(d) = \rho c \omega a(d), \]where \(a(d)\) is the amplitude at distance \(d\). Substitute \(a(d) = \frac{A R^2}{d}\) into the equation to find the pressure amplitude.

Key Concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion where an object oscillates back and forth through an equilibrium position. In this case, the sphere's radius oscillates between \(R - \Delta R\) and \(R + \Delta R\) following the sinusoidal pattern of SHM. The term "amplitude" \(A\) refers to the maximum displacement from the equilibrium, which is \(\Delta R\) in our scenario. The frequency of this oscillation is \(f\), and the angular frequency \(\omega\) is calculated as \(\omega = 2\pi f\). This angular frequency determines how quickly the sphere completes one cycle of motion.

• The equation \(R(t) = R + A \cos(\omega t)\) models the position of the sphere's surface at any time \(t\).
• The derivative \(v(t) = -A \omega \sin(\omega t)\) provides insight into the velocity of the sphere's surface, describing how fast the surface is moving at any point in time.

Understanding SHM is crucial because it forms the basis to analyze further concepts such as acoustic intensity and pressure variations, which are generated by the pulsating sphere.

Acoustic Intensity

Acoustic intensity is a measure of the energy carried by sound waves per unit area per unit time. In simpler terms, it tells us how "loud" or "strong" a sound is at a particular point. For the sphere, acoustic intensity \(I\) at the surface depends on the maximum velocity \(v_{max}\), the density of the air \(\rho\), and the speed of sound \(c\).
The speed of sound \(c\) in the air is calculated as \(c = \sqrt{\frac{B}{\rho}}\), where \(B\) is the bulk modulus, a property that describes the medium's resistance to compression. The formula for intensity becomes:
\[I = \frac{1}{2} \rho A^2 \omega^2 \sqrt{\frac{B}{\rho}}\]
This expression reveals how sound intensity increases with greater density \(\rho\), amplitude \(A\), and angular frequency \(\omega\). This specific relationship helps us understand how modifications in the sphere’s motion or the medium can affect sound production.

Sound Waves

Sound waves are pressure waves through a medium, like air, resulting from mechanical vibrations. When the pulsating sphere oscillates, it becomes a source of spherical sound waves that propagate through the air. These waves carry energy away from the sphere, and the acoustic power \(P\) illustrates the total energy radiated by these waves.
The power radiated by the sphere is determined by the intensity at its surface and the surface area itself:
\[P = 4 \pi R^2 I\]
Here, the incorporation of the surface area \(4 \pi R^2\) takes into account the entire area over which these waves spread. Sound intensity decreases as the distance from the source \(d\) increases, typical of spherical wave expansion whereby intensity \(I(d)\) is inversely proportional to \(d^2\).

• At greater distances, the amplitude \(a(d)\) of the waves reduces, described as \(\frac{A R^2}{d}\).
• Consequently, pressure amplitude \(p_0(d)\), the measure of pressure variation within the waves, can be calculated with the relation \(p_0(d) = \rho c \omega a(d)\).

Thus, understanding sound waves and their behavior is key to tackling problems related to acoustic intensity and power.