Problem 73 - Fundamentals of Physics

Question

At time \(t=0\) and at position \(x=0 \mathrm{~m}\) along a string, a traveling sinusoidal wave with an angular frequency of \(440 \mathrm{rad} / \mathrm{s}\) has displacement \(y=+4.5 \mathrm{~mm}\) and transverse velocity \(u=-0.75 \mathrm{~m} / \mathrm{s}\). If the wave has the general form \(y(x, t)=y_{m} \sin (k x-\omega t+\phi)\), what is phase constant \(\phi ?\)

Step-by-Step Solution

Answer

The phase constant \(\phi\) is approximately 1.212 radians.

Step 1: Write down the wave equation

The equation of the wave is given as \(y(x,t) = y_m \sin(kx - \omega t + \phi)\) where \(y_m\) is the amplitude, \(k\) is the wave number, \(\omega\) is the angular frequency, and \(\phi\) is the phase constant.

Step 2: Evaluate the given parameters

At \(t = 0\) and \(x = 0\), the displacement \(y = +4.5 \, \text{mm} = 0.0045 \, \text{m}\) and the transverse velocity \(u = -0.75 \, \text{m/s}\). The angular frequency \(\omega = 440 \, \text{rad/s}\).

Step 3: Relate displacement to the phase equation

At \(x = 0\) and \(t = 0\), the wave equation simplifies to \(y_m \sin(\phi) = 0.0045 \, \text{m}\).

Step 4: Derive the transverse velocity expression

The transverse velocity of the wave is given by differentiating the displacement: \(u(x, t) = \frac{\partial y}{\partial t} = -y_m \omega \cos(kx - \omega t + \phi)\). At \(t=0\) and \(x=0\), it simplifies to \(-y_m \omega \cos(\phi) = -0.75 \, \text{m/s}\).

Step 5: Solve for the phase constant \(\phi\)

From Steps 3 and 4:1. \(y_m \sin(\phi) = 0.0045\)2. \(-y_m \omega \cos(\phi) = -0.75\)Substitute \(\omega = 440\) to get \(y_m \cos(\phi) = \frac{0.75}{440}\).By dividing 1 by 2, using \(\tan(\phi) = \frac{\sin(\phi)}{\cos(\phi)}\), we have \(\tan(\phi) = 0.0045\times \frac{440}{0.75}\). Solve this to find \(\phi\).

Step 6: Calculate \(\tan(\phi)\) and solve for \(\phi\)

Compute \(\tan(\phi) = \frac{0.0045 \times 440}{0.75} = 2.64\). Use the inverse tangent function: \(\phi = \tan^{-1}(2.64)\). Calculate this to find \(\phi \approx 1.212\, \text{radians}\).

Key Concepts

Sinusoidal Waves

Sinusoidal waves are waves that have the shape of a sine or cosine function. They are important in wave mechanics because they provide a simple harmonic representation of oscillating phenomena. Imagine the wave as ripples on the surface of water, where each crest and trough follows a smooth, repetitive pattern.
Some key features of sinusoidal waves include:

• Amplitude (\(y_m \)): The maximum height of the wave crest or depth of the trough from the central axis. It measures how tall the wave is.
• Wavelength (\(\lambda \)): The distance between two crests or two troughs in the wave. It defines the length of one complete wave cycle.
• Frequency (\(f \)): The number of cycles that pass a specific point in one second.
• Wave speed (\(v \)): Calculated by the relation \(v = f\lambda\), representing how fast the wave propagates through a medium.
• Wave number (\(k\)): Defined as \(k = \frac{2\pi}{\lambda}\), it describes the number of wave cycles in a unit length.

This simple, oscillating shape makes sinusoidal waves a fantastic candidate for modeling various physical phenomena, from sound waves to electromagnetic waves, in a wide range of scientific fields.

Angular Frequency

Angular frequency is a measure of how quickly the wave oscillates in time. It is denoted by \(\omega \) and described in units of radians per second (\(\text{rad/s} \)). In wave mechanics, angular frequency connects the concept of a wave's frequency to its temporal oscillation. For a sinusoidal wave, it is related to the frequency (\(f\)) by the formula:\[\omega = 2\pi f\]This relationship shows how many radians a wave oscillates through in one second.
The angular frequency is integral in determining other wave properties:

• Higher angular frequencies indicate rapid oscillations, whereas lower angular frequencies suggest slower oscillations.
• It is a crucial variable in defining the **phase velocity** of a wave, expressed as the ratio of angular frequency to the wave number (\(v_p = \frac{\omega}{k}\)).
• For our exercise, we know \(\omega = 440 \, \text{rad/s}\), already hinting at a rapidly oscillating wave.

Understanding angular frequency helps us analyze how various waveforms behave in different settings, making it essential knowledge for tackling problems in physics and engineering.

Phase Constant

The phase constant (\(\phi \)) measures the initial angle of a sinusoidal wave at time \(t = 0\) and position \(x = 0\). It determines how much the wave is "shifted" horizontally along the wave cycle and plays an essential role in wave mechanics by helping describe the wave's initial conditions.
A few key points about the phase constant include:

• The value of \(\phi \) ranges typically from \(-\pi\) to \(\pi\) radians.
• A positive \(\phi \) value indicates a wave shifted to the left, while a negative \(\phi \) signifies a rightward shift.
• \(\phi \) is essential for determining the exact position of a wave at any point in space and time relative to its reference position, aiding in wave interference, beating, and other phenomena.

In our problem, by using both displacement and transverse velocity expressions, we calculated the phase constant as approximately \(1.212\) radians. This precise determination helps completely define the wave's behavior at its initial moment, crucial for understanding its dynamics in context.