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Chapter 18: Problem 67

Two waves are described by the wave functions $$ y_{1}(x, t)=5.0 \sin (2.0 x-10 t) $$ and $$ y_{2}(x, t)=10 \cos (2.0 x-10 t) $$ where \(y_{1}, y_{2},\) and \(x\) are in meters and \(t\) is in seconds. Show that the wave resulting from their superposition is also sinusoidal. Determine the amplitude and phase of this sinusoidal wave.

Short Answer

Expert verified
The resulting wave from superposition is also sinusoidal with amplitude approximately 11.18 meters and phase approximately -63.44 degrees.

Step by step solution

01

Understand Superposition

Superposition is the concept where the net displacement at a point is simply the vector sum of the displacements due to each individual wave. So adding the wave functions \(y_{1}(x, t)\) and \(y_{2}(x, t)\) will give the resulting wave function.
02

Sum of the waves

The resulting wave from the superposition of the given functions \(y_{1}(x, t)\) and \(y_{2}(x, t)\) is given by: \(y(x, t)=y_{1}(x, t) + y_{2}(x, t) = 5.0 \sin (2.0 x-10 t) + 10 \cos (2.0 x-10 t)\)
03

Trigonometric Conversion

We can convert the sum of the sine and cosine functions into a single sine function. This can be done using the following trigonometric identity: \(a \sin x + b \cos x = \sqrt{a^2 + b^2}\sin(x + \arctan(b/a))\). Applying the identity, \(y(x, t) = \sqrt{(5.0)^2 + (10)^2} \sin((2.0 x - 10t) + \arctan(-10/5))\).
04

Find the Amplitude and Phase

Now, we can identify the amplitude and phase of the resultant wave. In a sinusoidal function \(A \sin(x + \phi)\), the amplitude A is the peak value or the maximum absolute value of the function, and the phase shift is the horizontal shift of the sinusoidal function. From the derived equation, the amplitude A is \(\sqrt{(5.0)^2 + (10)^2}\) and the phase \(\phi\) is \(\arctan(-10/5)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wave Interference
Wave interference occurs when two or more waves overlap, resulting in a new wave pattern. This principle is a cornerstone in the field of physics, especially when discussing wave behavior in various mediums.

During interference, waves can constructively combine to produce a greater amplitude, or they can destructively combine, partially or completely canceling each other out. The specific outcome of wave interference is highly dependent on the phase relationship of the overlapping waves. If the waves meet in phase (their crests align), they will interfere constructively; if the waves are out of phase (crests align with troughs), they will interfere destructively.

In our problem, when the two waves described by their respective wave functions are combined, the resulting pattern is the product of their interference. The superposition principle facilitates the prediction of this pattern by algebraically adding the wave functions.
Trigonometric Identities
Trigonometric identities are equalities involving trigonometric functions that are true for all values of the variable for which the function is defined. These identities are indispensable tools for simplifying and solving complex trigonometric equations and expressions.

Common identities include Pythagorean identities, angle sum and difference identities, and double angle formulas. In the context of wave equations, they allow us to express a sum of sine and cosine functions as a single sinusoid. For instance, the identity used to combine our wave functions is derived from the angle sum identity, which is crucial for finding the amplitude and phase of the resultant wave.
Sinusoidal Wave Properties
Sinusoidal waves, such as sine and cosine functions, are the most fundamental periodic waveforms found in nature. These waves are characterized by their amplitude, wavelength, frequency, and phase.

The amplitude of a sinusoidal wave represents its maximum displacement from equilibrium, which determines the wave's power or intensity. Wavelength defines the spatial period of the wave鈥攖he distance over which the wave's shape repeats. Frequency measures how often the wave oscillates in a given time frame, often described in Hertz (Hz), which equals one oscillation per second. Lastly, phase describes the horizontal shift of the wave, and it is crucial when determining the interference between waves.

In superimposing waves, understanding these properties helps us predict the resulting wave's characteristics, allowing us to accurately describe the combined wave's amplitude and phase.
Amplitude and Phase of Waves
Amplitude and phase are key characteristics of sinusoidal waves that define their shape and position. The amplitude, the 'height' of the wave, indicates its strength or level of disturbance. The phase dictates where a wave starts in its cycle and is vital when combining waves, as it determines how the waves align and interfere with each other.

The amplitude of a combined wave is determined from the peak of the resultant wave and can be calculated using trigonometric identities, as seen in the exercise. While phase shift, often represented as a horizontal translation of the wave function, reflects how much a wave is shifted horizontally from a reference point. These properties together describe the resultant wave's configuration and are critical for understanding not just the wave's visual representation but also its physical implications in real-world scenarios.

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Most popular questions from this chapter

Standing Waves in Air Columns Note: Unless otherwise specified, assume that the speed of sound in air is \(343 \mathrm{m} / \mathrm{s}\) at \(20^{\circ} \mathrm{C},\) and is described by $$ v=(331 \mathrm{m} / \mathrm{s}) \sqrt{1+\frac{T_{\mathrm{C}}}{273^{\circ}}} $$ at any Celsius temperature \(T_{\mathrm{C}}\). The overall length of a piccolo is \(32.0 \mathrm{cm} .\) The resonating air column vibrates as in a pipe open at both ends. (a) Find the frequency of the lowest note that a piccolo can play, assuming that the speed of sound in air is \(340 \mathrm{m} / \mathrm{s} .\) (b) Opening holes in the side effectively shortens the length of the resonant column. If the highest note a piccolo can sound is \(4000 \mathrm{Hz}\), find the distance between adjacent antinodes for this mode of vibration.

With a particular fingering, a flute plays a note with frequency \(880 \mathrm{Hz}\) at \(20.0^{\circ} \mathrm{C} .\) The flute is open at both ends. (a) Find the air column length. (b) Find the frequency it produces at the beginning of the half-time performance at a late-season American football game, when the ambient temperature is \(-5.00^{\circ} \mathrm{C}\) and the musician has not had a chance to warm up the flute.

A loudspeaker at the front of a room and an identical loudspeaker at the rear of the room are being driven by the same oscillator at \(456 \mathrm{Hz}\). A student walks at a uniform rate of \(1.50 \mathrm{m} / \mathrm{s}\) along the length of the room. She hears a single tone, repeatedly becoming louder and softer. (a) Model these variations as beats between the Doppler-shifted sounds the student receives. Calculate the number of beats the student hears each second. (b) What If? Model the two speakers as producing a standing wave in the room and the student as walking between antinodes. Calculate the number of intensity maxima the student hears each second.

When beats occur at a rate higher than about 20 per second, they are not heard individually but rather as a steady hum, called a combination tone. The player of a typical pipe organ can press a single key and make the organ produce sound with different fundamental frequencies. She can select and pull out different stops to make the same key for the note \(\mathrm{C}\) produce sound at the following frequencies: \(65.4 \mathrm{Hz}\) from a so-called eight-foot pipe; \(2 \times 65.4=\) \(131 \mathrm{Hz}\) from a four-foot pipe; \(3 \times 65.4=196 \mathrm{Hz}\) from a two-and-two-thirds-foot pipe; \(4 \times 65.4=262 \mathrm{Hz}\) from a two-foot pipe; or any combination of these. With notes at low frequencies, she obtains sound with the richest quality by pulling out all the stops. When an air leak develops in one of the pipes, that pipe cannot be used. If a leak occurs in an eight-foot pipe, playing a combination of other pipes can create the sensation of sound at the frequency that the eight-foot pipe would produce. Which sets of stops, among those listed, could be pulled out to do this?

A student holds a tuning fork oscillating at \(256 \mathrm{Hz}\). He walks toward a wall at a constant speed of \(1.33 \mathrm{m} / \mathrm{s}\) (a) What beat frequency does he observe between the tuning fork and its echo? (b) How fast must he walk away from the wall to observe a beat frequency of \(5.00 \mathrm{Hz} ?\)
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