Question

A traveling wave propagating on a string is described by the following equation: $$ y(x, t)=(5.00 \mathrm{~mm}) \sin \left(\left(157.08 \mathrm{~m}^{-1}\right) x-\left(314.16 \mathrm{~s}^{-1}\right) t+0.7854\right) $$ a) Determine the minimum separation, \(\Delta x_{\text {min }}\), between two points on the string that oscillate in perfect opposition of phases (move in opposite directions at all times). b) Determine the minimum separation, \(\Delta x_{A B}\), between two points \(A\) and \(B\) on the string, if point \(B\) oscillates with a phase difference of \(0.7854 \mathrm{rad}\) compared to point \(A\). c) Find the number of crests of the wave that pass through point \(A\) in a time interval \(\Delta t=10.0 \mathrm{~s}\) and the number of troughs that pass through point \(B\) in the same interval.

Short answer

Answer: a) The minimum separation for perfect opposition is Δ𝑥min = 0.02 m. b) The minimum separation for a phase difference of 0.7854 rad is Δ𝑥AB = 0.005 m. c) The number of crests passing through point A and troughs passing through point B is 500 each, in the given time interval.

Step-by-step solution

Determine the wavelength

The wave equation is given as: $$ y(x, t) = (5.00 \ \text{mm}) \sin\left((157.08 \ \text{m}^{-1}) x - (314.16 \ \text{s}^{-1}) t + 0.7854\right) $$ The wave number (the number of waves per unit length) equals: $$ k = 157.08 \ \text{m}^{-1} $$ We can determine the wavelength by using the following relation: $$ \lambda=\frac{2 \pi}{k} $$ Calculating the wavelength: $$ \lambda=\frac{2 \pi}{157.08 \ \text{m}^{-1}} \approx 0.04 \ \text{m} $$

Calculate the separations for perfect opposition and phase difference

To make the motion of two points in perfect opposition, they must differ by half the wavelength. This is because when one point is at the crest of the wave, the other point must be at the trough of the wave. Therefore, $$\Delta x_{\text{min}} = \frac{1}{2} \lambda$$ Calculate the minimum separation for perfect opposition: $$ \Delta x_{\text{min}} = \frac{1}{2}(0.04 \ \text{m}) = 0.02 \ \text{m} $$ For two points with a phase difference of 0.7854 rad, we need to find the value of \(\Delta x\). This can be done by equating the phase difference to the wave number multiplied by the required distance. $$ 0.7854 \ \text{rad} = (157.08 \ \text{m}^{-1}) \Delta x_{AB} $$ Now solve for \(\Delta x_{AB}\): $$ \Delta x_{AB}=\frac{0.7854 \ \text{rad}}{157.08 \ \text{m}^{-1}} \approx 0.005 \ \text{m} $$

Find the number of crests and troughs

To find the number of crests and troughs, we first need to find the frequency (\(f\)) using the equation $$f = \frac{\omega}{2\pi}$$ Where \(\omega = 314.16 \ \text{s}^{-1}\) is the angular frequency. Calculate the frequency: $$ f = \frac{314.16 \ \text{s}^{-1}}{2\pi} \approx 50 \ \text{Hz} $$ As given in the problem, the time interval is \(\Delta t = 10\ \text{s}\). Since the frequency is the number of oscillations per second, we can determine the number of crests and troughs passing through point A and point B by multiplying the frequency with the given time interval. $$ \text{Number of crests at A} = f \cdot \Delta t = 50 \ \text{Hz} \cdot 10\ \text{s} = 500 \ \text{crests} $$ Since A and B oscillate with a phase difference of 0.7854 rad, the number of troughs passing through point B is equal to the number of crests passing through point A. $$ \text{Number of troughs at B} = 500 \ \text{troughs} $$ Now, we have the answers: a) The minimum separation for perfect opposition is \(\Delta x_{\text {min}} = 0.02 \ \text{m}\) b) The minimum separation for a phase difference of 0.7854 rad is \(\Delta x_{AB}=0.005 \ \text{m}\) c) The number of crests passing through point A is 500 and the number of troughs passing through point B is also 500.

Key concepts

Wave Propagation

Wave propagation refers to the movement through space and time of waves, such as those rippling across the surface of a pond or traveling through a guitar string. When we speak of a traveling wave, we are describing the observation that the wave's peaks and troughs move through space over time, an example of which can be seen in the equation presented in the textbook exercise.

The wave equation for a traveling wave on a string, like the one in our exercise \( y(x, t) = (5.00 \, \text{mm}) \sin((157.08 \, \text{m}^{-1}) x - (314.16 \, \text{s}^{-1}) t + 0.7854) \) encapsulates all pertinent information about the wave's motion, including its amplitude, frequency, and phase. By understanding this equation, one can predict the position of any point on the string at any given time. The speed of a wave is determined by the medium through which it travels, the tension in the string, and its mass per unit length. Consequently, these factors also affect the wave's frequency and wavelength.

Oscillations and Wave Motion

Oscillations in a wave refer to the repetitive up and down (or side to side) motion seen in each point of the medium through which the wave moves. As the wave propagates, each individual particle in the medium completes a cycle of motion, returning to its initial position after a certain period. The relationship between oscillation and wave motion becomes clear when examining the oscillating particles in a medium over time, which collectively manifest as a traveling wave.

In our exercise, each point on the string undergoes periodic motion, described mathematically by the sinusoidal function in the wave equation. The wave's amplitude, the maximum displacement of a point from its rest position, is indicated by the 5.00 mm term, reflecting the intensity of the oscillation at any given point. Oscillation frequency, indicating how often the point returns to its initial position, ties in with the number of crests or troughs passing a fixed point in a given time frame. As calculated in the solution, the frequency of the oscillations for the given wave is approximately 50 Hz, meaning that each point on the string oscillates back and forth 50 times every second.

Phase Difference

The concept of phase difference is crucial in understanding how different points along a wave move relative to each other. Phase difference refers to the difference in the phase of oscillations of two points, which determines whether they reach the crests and troughs of the wave at the same time or not. A phase difference of 0 radians or a multiple of \(2\pi\) radians means the points are oscillating in unison, whereas a phase difference of \(\pi\) radians indicates that they are in perfect opposition, as one point is at a crest while the other is at a trough.

In the solved exercise, the phase difference is used to calculate the minimum separation between two points on the string. The separation for perfect opposition, \(\Delta x_{\text{min}} = 0.02 \, \text{m}\), indicates that this is the shortest distance at which two points will continually move in opposite directions. When a specific phase difference, such as 0.7854 rad (\(\frac{\pi}{4}\) radians), is introduced between two points, the separation changes accordingly. Knowledge of the phase difference allows for precise control over the motion and interaction of different points along a traveling wave.