Problem 43 - University Physics with Modern Physics

Question

Students in a lab produce standing waves on stretched strings connected to vibration generators. One such wave is described by the wave function \(y(x, t)=(2.00 \mathrm{~cm}) \sin \left[\left(20.0 \mathrm{~m}^{-1}\right) x\right] \cos \left[\left(150 . \mathrm{s}^{-1}\right) t\right],\) where \(y\) is the transverse displacement of the string, \(x\) is the position along the string, and \(t\) is time. Rewrite this wave function in the form for a right- moving and a left-moving wave: \(y(x, t)=\) \(f(x-v t)+g(x+v t)\); that is, find the functions \(f\) and \(g\) and the speed, \(v\)

Step-by-Step Solution

Answer

Question: Rewrite the given wave function as a sum of a right-moving wave function f(x-vt) and a left-moving wave function g(x+vt), and find the wave speed v. The given wave function: \(y(x, t) = (2.00 \mathrm{~cm}) \sin \left[\left(20.0 \mathrm{~m}^{-1}\right) x\right] \cos \left[\left(150 . \mathrm{s}^{-1}\right) t\right]\). Answer: The rewritten wave function is: \(y(x, t) = (1.00\mathrm{~cm})\sin \left(\left(20.0 \mathrm{~m}^{-1}\right)x - \left(150\mathrm{s}^{-1}\right)t\right) + (1.00\mathrm{~cm})\sin \left(\left(20.0 \mathrm{~m}^{-1}\right)x + \left(150\mathrm{s}^{-1}\right)t\right)\) The wave speed v is \(7.50\mathrm{~m/s}\).

Step 1: Identifying the wave function

The given wave function is: \(y(x, t) = (2.00 \mathrm{~cm}) \sin \left[\left(20.0 \mathrm{~m}^{-1}\right) x\right] \cos \left[\left(150 . \mathrm{s}^{-1}\right) t\right]\)

Step 2: Exprimenting with trigonometric identities

In order to express the given wave function as a sum of right-moving and left-moving waves, let's experiment with trigonometric identities. Recall the trigonometric identity for the product of sine and cosine: \(\sin(A) \cos(B) = \frac{1}{2} \left[\sin(A+B) + \sin(A-B)\right]\) Here, A can be considered as \((20.0\mathrm{~m}^{-1})x\) and B can be considered as \((150\mathrm{s}^{-1})t\). Applying this identity to the given wave function, we get: \(y(x, t) = (2.00\mathrm{~cm}) \cdot \frac{1}{2}\left[\sin \left(\left(20.0 \mathrm{~m}^{-1}\right)x + \left(150\mathrm{s}^{-1}\right)t\right) + \sin \left(\left(20.0 \mathrm{~m}^{-1}\right)x - \left(150\mathrm{s}^{-1}\right)t\right)\right]\)

Step 3: Writing the wave function as a sum of right-moving and left-moving waves

Now, we can rewrite the wave function as a sum of a right-moving wave function \(f(x-vt)\) and a left-moving wave function \(g(x+vt)\): \(y(x, t) = (1.00\mathrm{~cm})\sin \left(\left(20.0 \mathrm{~m}^{-1}\right)x + \left(150\mathrm{s}^{-1}\right)t\right) + (1.00\mathrm{~cm})\sin \left(\left(20.0 \mathrm{~m}^{-1}\right)x - \left(150\mathrm{s}^{-1}\right)t\right)\) Comparing this expression with the desired form \(y(x, t) = f(x-vt) + g(x+vt)\), we can identify the functions f and g, and the wave speed v:

Step 4: Identifying the functions f and g

From the comparison, the right-moving wave function \(f(x-vt)\) and the left-moving wave function \(g(x+vt)\) are: \(f(x-vt) = (1.00\mathrm{~cm})\sin \left(\left(20.0 \mathrm{~m}^{-1}\right)x - \left(150\mathrm{s}^{-1}\right)t\right)\) \(g(x+vt) = (1.00\mathrm{~cm})\sin \left(\left(20.0 \mathrm{~m}^{-1}\right)x + \left(150\mathrm{s}^{-1}\right)t\right)\)

Step 5: Identifying the wave speed v

From the expressions of f and g, the wave speed v is given by the ratio of the angular frequency divided by the wavenumber. Let's find v: \(ω = 150\mathrm{s}^{-1}\) \(k = 20.0\mathrm{~m}^{-1}\) Now, using the relation \(v = \frac{ω}{k}\), we can find the wave speed v: \(v = \frac{150\mathrm{s}^{-1}}{20.0\mathrm{~m}^{-1}} = 7.50\mathrm{~m/s}\) Finally, the functions f, g and the speed v are: \(f(x-vt) = (1.00\mathrm{~cm})\sin \left(\left(20.0 \mathrm{~m}^{-1}\right)x - \left(150\mathrm{s}^{-1}\right)t\right)\) \(g(x+vt) = (1.00\mathrm{~cm})\sin \left(\left(20.0 \mathrm{~m}^{-1}\right)x + \left(150\mathrm{s}^{-1}\right)t\right)\) \(v = 7.50\mathrm{~m/s}\)

Key Concepts

Wave Function

A wave function mathematically describes the evolution of a wave in space and time. In the example provided, the wave function \(y(x, t) = (2.00 \mathrm{~cm}) \sin \left[\left(20.0 \mathrm{~m}^{-1}\right) x\right] \cos \left[\left(150 . \mathrm{s}^{-1}\right) t\right]\) includes parameters that represent the transverse displacement (the height of a point on the wave from its rest position), and its dependence on both time (t) and position (x) along the string.

Understanding the wave function is crucial for analyzing the properties of waves, including their speed, frequency, and wavelength. In the context of standing waves, like those created in a vibration experiment with strings, the wave function helps to express the combined motion of two waves moving in opposite directions, which can be depicted as the sum of right-moving and left-moving waves.

Trigonometric Identities

Trigonometric identities are mathematical relationships that express one trigonometric function in terms of others. These can simplify complex equations and are particularly useful in wave analysis. In this specific exercise, the product-to-sum identity \(\sin(A) \cos(B) = \frac{1}{2} \left[\sin(A+B) + \sin(A-B)\right]\) was used.

By substituting the appropriate parts of the wave function with \(A\) and \(B\), the original wave function was transformed to highlight the individual components of two sinusoidal waves moving in opposite directions. This transformation is central to revealing the two functions \(f\) and \(g\) that represent the right-moving and left-moving waves respectively.

Wave Speed

Wave speed is a fundamental characteristic of a wave, defining how fast the wave propagates through a medium. For a standing wave on a string, like in our example, the wave speed \(v\) can be determined by analyzing the wave function. It is the rate at which the wave's crests and troughs move along the string and is derived using the relationship between angular frequency \(\omega\) and wavenumber \(k\), given by \(v = \frac{\omega}{k}\).

The speed of the wave tells us how quickly the pattern of disturbances created by the wave will travel along the string. In the context of standing waves, the wave speed also conveys how quickly the nodes and antinodes appear to oscillate, although visibly they remain in the same position, a characteristic feature of standing waves.

Transverse Displacement

Transverse displacement in the context of waves refers to the deviation of points on a wave from their original, undisturbed positions, measured perpendicular to the direction of wave travel. In other words, it's the 'height' of the wave at any given point. The transverse displacement at any point in time and at any location along the medium is represented by the wave function. Higher displacement corresponds to regions of greater wave amplitude—a concept crucially important for understanding phenomena such as interference and resonance in standing waves.

For students studying standing waves, visualizing transverse displacement as a function of position and time can greatly aid in comprehending the otherwise abstract concept of waveform evolution and energy transfer through a medium.

Angular Frequency

Angular frequency \(\omega\), measured in radians per second, describes how rapidly a wave oscillates in time. It is directly related to the frequency of the wave, with a higher angular frequency indicating a higher number of oscillations per second. In the equation provided, \(\omega = 150 \mathrm{s}^{-1}\) represents the wave’s temporal rate of change.

Understanding angular frequency is important for analyzing wave behaviors, such as those in standing waves, where two waves of the same frequency and amplitude travel in opposite directions, creating points of constructive and destructive interference, namely the nodes (points of no movement) and antinodes (points of maximum movement).

Wavenumber

Wavenumber \(k\) is a measure of the number of wave cycles per unit distance and is directly related to the wavelength, the distance over which the wave's shape repeats. It is expressed in our wave function as \(20.0 \mathrm{~m}^{-1}\) and dictates the spatial frequency of the wave along the string. The wavenumber provides insight into the structure of the wave in space.

Understanding wavenumber is critical when discussing wave properties such as phase velocity and when dealing with concepts like the dispersion relation, which shows the relationship between wavenumber and angular frequency for waves traveling through different media, which is particularly important for waves in more complex systems beyond simple strings.