AP Physics 1 Exam Question:

A sound wave is propagating through air. The region of compression and rarefaction in the wave leads to a constant pressure variation with time. Given the equation of the wave is given by:

Where:

• $P$ is the pressure variation from ambient pressure (in Pascals),
• $x$ is the displacement from equilibrium position (in meters),
• $t$ is time (in seconds),
• $k$ is the wave number (in inverse meters),
• $\omega$ is the angular frequency (in radians per second).

a) Identify the quantities that determine the properties of this sound wave.

b) If the wave number $k = 4.0 \, \text{m}^{-1}$ and the angular frequency $\omega = 10 \, \text{rad/s}$, determine the period, frequency, and velocity of the sound wave.

c) Explain how the amplitude of the wave affects the energy carried by the wave.

Answer:

a) The properties of this sound wave are determined by the following quantities:

• Wave number ($k$): Determines the spatial frequency of the wave and describes the rate at which the wave oscillates through space. It is inversely proportional to the wavelength.
• Angular frequency ($\omega$): Determines the temporal frequency of the wave and describes the rate at which the wave oscillates in time. It is directly proportional to the period and angular velocity.
• Pressure variation amplitude ($A$): Determines the peak value of the pressure variation from the ambient pressure, representing the maximum displacement of air particles from their equilibrium positions.
• Position ($x$): Represents the displacement from the equilibrium position and determines the location along the wave.
• Time ($t$): Represents the instant in time during which the wave is being observed.

b) Given:

• Wave number $k = 4.0 \, \text{m}^{-1}$
• Angular frequency $\omega = 10 \, \text{rad/s}$

To determine the period ($T$), frequency ($f$), and velocity ($v$) of the sound wave, we can use the following relationships:

Substituting the given values, we can evaluate the equations as follows:

The period of the wave is $\frac{\pi}{5}$ seconds, the frequency is $\frac{5}{\pi}$ Hertz, and the velocity of the wave is $2.5$ m/s.

c) The amplitude of the wave ($A$) affects the energy carried by the wave. The energy carried by a wave is directly proportional to the square of the amplitude. Mathematically, the energy ($E$) carried by a wave is given by:

Where $\rho$ is the density of the medium through which the wave is propagating. As the amplitude of the wave increases, the energy carried by the wave also increases, given that the other quantities (density and velocity) remain constant. This is evident in the equation, as the amplitude is squared. Therefore, higher amplitude waves have a greater energy content.