91Ó°ÊÓ

Analyzing wave functions is essential in understanding the behavior of waves, particularly standing waves, which are formed by the superposition of two traveling waves. The wave function given in the exercise,
\( y(x, t) = (4.44 \, \text{mm}) \sin [(32.5 \, \text{rad/m}) x] \sin [(754 \, \text{rad/s}) t] \),
provides all the necessary information to determine the properties of the individual traveling waves that combine to create the standing wave. By inspecting the wave function, we can grasp that the sine components refer to the spatial and temporal oscillations, respectively.

Interpreting the Wave Function

The spatial part, \( \sin [(32.5 \, \text{rad/m}) x] \), indicates how the wave varies along the x-axis, while the temporal part, \( \sin [(754 \, \text{rad/s}) t] \), shows how the wave oscillates over time. Together, they form a pattern that does not travel through space but instead oscillates in a fixed position, characteristic of standing waves. For a student striving to visualize this concept, it might be helpful to picture the wave as a vibrating string fixed at both ends, with certain points, called nodes, remaining stationary, while antinodes achieve the maximum amplitude of vibration.

Traveling wave equations describe the motion and form of waves as they propagate through space. In the context of the exercise, the two traveling waves that combine to form the standing wave have specific equations based on the given wave function.

Constructing the Traveling Waves

To construct the equations of the individual traveling waves, we utilize the principle that standing waves are the result of the interference of two traveling waves moving in opposite directions. These traveling waves have wave functions expressed as:

• \( y_1(x,t) = A \sin(kx - \omega t) \)
• \( y_2(x,t) = A \sin(kx + \omega t) \)

Where \( A \) is the amplitude, \( k \) the wave number related to wavelength, and \( \omega \) the angular frequency related to the frequency of the waves. The negative sign in the first equation indicates that the wave travels in the positive x-direction, while the positive sign in the second equation implies that the wave travels in the negative x-direction. With this understanding, one can tackle a range of wave-related problems and decipher the behavior of moving wave phenomena.

Understanding how to calculate the wavelength of a wave is crucial in physics and engineering. The wavelength is the physical distance between successive points of the same phase in the wave, such as peak to peak or trough to trough.

Deducing Wavelength from the Wave Number

The step-by-step solution demonstrates that the wave number \( k \) contains the information needed for calculating the wavelength \( \lambda \). The relationship between the two is given by \( k = \frac{2\pi}{\lambda} \). By rearranging this equation and substituting the given wave number, we find:

• \( \lambda = \frac{2\pi}{k} \)

In the context of our exercise, with \( k = 32.5 \, \text{rad/m} \), the computed wavelength is \( \lambda = 0.193 \, \text{m} \). Knowing the wavelength is essential for applications involving tuning musical instruments or designing antennas where specific wave properties are needed.

Frequency is a key concept when dealing with waves as it defines the number of oscillations that occur in one second, and it's inversely related to the period of the wave. In a practical context, frequency can affect everything from the pitch of a sound to the color of light perceived by our eyes.

Calculating Frequency from Angular Frequency

The angular frequency \( \omega \) provided in the wave function is directly proportional to the frequency \( f \), with the relationship given by \( \omega = 2\pi f \). By solving for \( f \) using the value of \( \omega \) from the problem, we establish the frequency of the wave:

• \( f = \frac{\omega}{2\pi} \)

For our exercise, with \( \omega = 754 \, \text{rad/s} \), the frequency is \( f = 120 \, \text{Hz} \). Identifying the frequency is crucial for many applications such as diagnosing medical conditions with ultrasound, broadcasting radio signals, or even simply setting up a Wi-Fi network at home.