91Ó°ÊÓ

A wave function is a mathematical description of a wave. It depicts how the wave behaves at different positions and times. For a standing wave, the wave function can be expressed in terms of sine and cosine functions. In this exercise, the wave function is given as:
\( y(x, t) = 4.44\, \text{mm} \sin [(32.5\, \text{rad/m}) x] \sin [(754\, \text{rad/s}) t] \).
This particular form is typical for standing waves, representing the combination of two traveling waves moving in opposite directions. Standing waves occur when waves are confined within a certain space, such as in a string fixed at both ends. Note that the standing wave's amplitude is

• Maximum at antinodes (peaks)
• Zero at nodes (no displacement)

In this wave function, parameters such as wave number and angular frequency are visible, and they help to derive other properties of the wave.

Wavelength is the distance over which the wave's shape repeats. In essence, it's the length of one complete wave cycle. For the given wave function, the wave number \( k \) is given as \( 32.5 \, \text{rad/m} \). The relationship between wavelength \( \lambda \) and wave number is given by:
\[ k = \frac{2\pi}{\lambda} \]
From the wave function, we can solve for the wavelength:
\[ \lambda = \frac{2\pi}{32.5} \, \text{m} \approx 0.193 \, \text{m} \]
Thus, the given wavelength is approximately 0.193 meters. This parameter is crucial as it describes the physical length of each wave cycle in the standing wave.

Frequency refers to how many wave cycles pass through a particular point per second. It's measured in Hertz (Hz). In the wave function provided, the angular frequency \( \omega \) is given as \( 754 \, \text{rad/s} \). The connection between angular frequency and linear frequency \( f \) is:
\[ \omega = 2\pi f \]
Consequently, the frequency is:
\[ f = \frac{\omega}{2\pi} = \frac{754}{2\pi} \, \text{Hz} \approx 120 \, \text{Hz} \]
At approximately 120 Hz, this frequency indicates how many times the pattern of the wave repeats itself each second. A higher frequency means the wave cycles repeat more frequently over time, which is key for understanding the wave's dynamics.

In wave systems, especially in physics involving strings or air columns, the harmonic number \( n \) denotes the mode of vibration. It tells us how many segments or loops are formed within the boundary conditions of the system.
The relation \( k_n = \frac{n\pi}{L} \) can help determine harmonic number if the system's length \( L \) is known. Here, \( k \) is provided as \( 32.5 \, \text{rad/m} \), but without the system's length \( L \), the harmonic number remains indeterminate.

• If we knew \( L \), we could solve \( n = \frac{kL}{\pi} \).
• Thus, the harmonic number indicates the number of nodes or loop segments present.
• Harmonics are integral for analyzing musical instruments, wave resonance, and various physical systems.

Without \( L \), precise identification of the harmonic is not possible in this problem.