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 wave moving in the positive \(x\) -direction and a wave moving in the negative \(x\) -direction: \(y(x, t)=f(x-v t)+g(x+v t) ;\) that is, find the functions \(f\) and \(g\) and the speed, \(v\)

Short answer

Question: Determine the two wave functions representing waves moving in the positive x-direction and the negative x-direction, and find the wave speed of the given standing wave. Answer: The two wave functions are f(x-vt) = (2.00 cm) * (1/2) * sin[(20.0 m⁻¹)(x - (150 s⁻¹)t)] and g(x+vt) = (2.00 cm) * (1/2) * sin[(20.0 m⁻¹)(x + (150 s⁻¹)t)]. The wave speed is 150 s⁻¹.

Step-by-step solution

Rewrite the given wave function

We can make use of the identity \(\sin a \cos b = \frac{1}{2}(\sin (a-b)+\sin (a+b))\) to rewrite the wave function as follows: $y(x, t)=(2.00 \mathrm{~cm}) \sin \left[\left(20.0 \mathrm{~m}^{-1}\right) x\right] \cos \left[\left(150 . \mathrm{s}^{-1}\right) t\right] = (2.00 \mathrm{~cm})\left[\frac{1}{2} \left(\sin \left[\left(20.0 \mathrm{~m}^{-1}\right) (x - (150 \mathrm{s}^{-1}) t)\right] + \sin \left[\left(20.0 \mathrm{~m}^{-1}\right) (x + (150 \mathrm{s}^{-1}) t)\right]\right)\right]$

Rewrite the wave function in the required format

Now, rewrite the wave function in the required format \(y(x, t) = f(x-vt) + g(x+vt)\): \(y(x, t) = (2.00 \mathrm{~cm})\frac{1}{2}\left( \sin \left[\left(20.0 \mathrm{~m}^{-1}\right) (x - (150 \mathrm{s}^{-1}) t)\right] + \sin \left[\left(20.0 \mathrm{~m}^{-1}\right) (x + (150 \mathrm{s}^{-1}) t)\right]\right)\)

Identify f(x-vt), g(x+vt) and v

By comparing the rewritten wave function with the required format, we can identify the function f(x-vt), g(x+vt) and the speed v: \(f(x-vt) = (2.00 \mathrm{~cm})\frac{1}{2}\sin\left[\left(20.0 \mathrm{~m}^{-1}\right)(x - (150 \mathrm{s}^{-1})t)\right]\) \(g(x+vt) = (2.00 \mathrm{~cm})\frac{1}{2}\sin\left[\left(20.0 \mathrm{~m}^{-1}\right)(x + (150 \mathrm{s}^{-1})t)\right]\) \(v = 150 \mathrm{s}^{-1}\) Thus, we have found that the two wave functions \(f(x-vt)\) and \(g(x+vt)\) representing waves moving in the positive \(x\)-direction and the negative \(x\)-direction, respectively, are: \(f(x-vt) = (2.00 \mathrm{~cm})\frac{1}{2}\sin\left[\left(20.0 \mathrm{~m}^{-1}\right)(x - (150 \mathrm{s}^{-1})t)\right]\) \(g(x+vt) = (2.00 \mathrm{~cm})\frac{1}{2}\sin\left[\left(20.0 \mathrm{~m}^{-1}\right)(x + (150 \mathrm{s}^{-1})t)\right]\) And the wave speed, \(v\), is \(150 \mathrm{s}^{-1}\).

Key concepts

Wave Function

In the realm of physics, wave function is a mathematical description that defines the wave-like nature of particles and how they evolve over time within a system. When it comes to standing waves, particularly on a string, the wave function precisely describes the displacement of any point on the string at a given time.

The exercise provided presents a complex wave function for a standing wave on a string. It is crucial for students to understand how to manipulate this function to gain insights into wave properties. The use of trigonometric identities allows us to separate this complex wave function into components that represent waves moving in opposite directions along the string. This decomposition into simpler forms is a powerful concept because it enables students to analyze each moving wave individually.

In this case, the original wave function's transformation into the sum of two sine waves creates an opportunity to understand how waves can interfere and form standing waves, which are essential phenomena in fields like acoustics and instrument design.

Transverse Displacement

The term transverse displacement in wave mechanics refers to the perpendicular motion of elements of the medium (e.g., a string) relative to the direction of wave propagation. It's a measure of how far points on the string move from their equilibrium position as the wave passes. This concept is a cornerstone in understanding how waves transport energy across a medium without transferring mass.

In the standing wave equation from the exercise, the transverse displacement is represented by the amplitude of the sine functions. By exploring this equation, students can visualize the wave's peaks and troughs and understand how the amplitude affects the intensity of the wave's motion.

This is crucial in practical applications, such as engineering, where the transverse displacement can affect the vibration patterns and stability of structures, making the concept significant for safety and design considerations.

Wave Speed

Another fundamental concept is wave speed, which represents how fast a wave moves through its medium. For a fixed frequency, the wave speed can significantly impact both the wavelength and the standing wave patterns on the medium. It is calculated as the product of the wavelength and frequency.

The wave speed in the exercise is derived from the transformed wave function, indicating the speed at which the waves travel along the string in opposite directions. Understanding the determination of wave speed helps students to grasp concepts such as wave interference and resonance, which play essential roles in systems like musical instruments, and scientific applications like medical imaging.

Recognizing the relation between wave speed, frequency, and wavelength is vital for any student studying wave mechanics and is widely applicable across various forms of wave phenomena including sound waves, electromagnetic waves, and water waves.