Problem 83 - Fundamentals of Physics

Question

A sinusoidal transverse wave of amplitude \(y_{m}\) and wavelength \(\lambda\) travels on a stretched cord. (a) Find the ratio of the maximum particle speed (the speed with which a single particle in the cord moves transverse to the wave) to the wave speed. (b) Does this ratio depend on the material of which the cord is made?

Step-by-Step Solution

Answer

(a) \( \frac{2 \pi y_m}{\lambda} \); (b) No, it does not depend on the material.

Step 1: Understanding Particle Speed and Wave Function

The transverse displacement of a point in a sinusoidal wave can be expressed as \( y(x, t) = y_m \sin(kx - \omega t) \), where \( y_m \) is the amplitude, \( k = \frac{2 \pi}{\lambda} \) is the wave number, and \( \omega \) is the angular frequency. The maximum particle speed is the maximum transverse velocity of the point.

Step 2: Calculating Maximum Particle Speed

To find the velocity of a particle at any point, we take the derivative of the displacement with respect to time: \( v(x, t) = \frac{\partial y}{\partial t} = -y_m \omega \cos(kx - \omega t) \). The maximum particle speed occurs when \( \cos(kx - \omega t) = \pm 1 \), giving \( v_{max} = y_m \omega \).

Step 3: Expressing Wave Speed

The wave speed \( v \) is given by the relationship between angular frequency and wave number: \( v = \frac{\omega}{k} = \frac{\omega \lambda}{2 \pi} \).

Step 4: Ratio of Maximum Particle Speed to Wave Speed

The ratio of the maximum particle speed to the wave speed is given by \( \frac{v_{max}}{v} = \frac{y_m \omega}{\frac{\omega \lambda}{2 \pi}} = \frac{2 \pi y_m}{\lambda} \).

Step 5: Analyzing Dependence on Material

The ratio \( \frac{2 \pi y_m}{\lambda} \) involves only the amplitude and wavelength. Thus, it does not depend on the material of the cord, as these variables are not related to material properties.

Key Concepts

Transverse Wave

A transverse wave is a type of wave where the motion of the medium is perpendicular to the direction of propagation of the wave. Picture a wave traveling along a stretched string. As the wave moves from left to right, each particle of the string moves up and down in a direction orthogonal to the wave's travel path. This creates a "wave" shape that's familiar to us when we think of ocean waves.

In a sinusoidal transverse wave, like the one described in the exercise, the displacement of the wave can be represented mathematically, which allows us to use equations to determine other properties like speed and amplitude. Understanding these different properties helps us to comprehend how energy travels through the wave.

Particle Speed

Particle speed in a wave context refers to how fast a single particle in the medium is moving. In the transverse wave case, this speed is transverse or perpendicular to the wave's direction.

To calculate the maximum speed of these particles, we use the derivative of the wave function concerning time. This derivative gives us a formula: \( v_{max} = y_m \omega \), where \( y_m \) is the amplitude, and \( \omega \) represents the angular frequency. The maximum speed highlights the fastest rate at which particles in the material can oscillate as the wave passes.

The speed of individual particles varies throughout their cycle, but this formula helps find the peak value.

Wave Speed

Wave speed is the speed at which the wave travels through the medium. For our transverse wave example, this is the speed at which the wave's crests and troughs move down the string.

The wave speed can be determined with the equation \( v = \frac{\omega}{k} \). In this formula, \( k \) is the wave number, and \( \omega \) is the angular frequency. Through further simplification and understanding relationships in wave mechanics, we can express wave speed as another equivalent equation: \( v = \frac{\omega \lambda}{2\pi} \).

• Wave speed and particle speed are different. While wave speed refers to the entire wave's travel speed, particle speed is specific to points in the medium oscillating with the wave.
• A critical insight from this solution is identifying that the wave speed does not rely on the material properties, such as tension or mass density, but rather on its frequency and wavelength.

Amplitude

Amplitude is the measure of how much energy is carried by the wave, reflected in the height of its crests. In our mathematical representation, amplitude is denoted by \( y_m \).

A wave with higher amplitude means that individual particles within the medium travel further away from their resting position when the wave passes. This can also be thought of as the wave's strength.

Understanding amplitude is crucial because it plays a role in determining the maximum speed of particles in the wave, as shown in how it's directly proportional to the maximum transverse speed formula \( v_{max} = y_m \omega \).

Greater amplitude suggests more energy is being transferred, whether the wave is in a rope, on water, or even sound waves through the air.

Wavelength

Wavelength is the distance over which the wave's shape repeats. It's often denoted by \( \lambda \). Imagine this as the distance from one wave crest to the next.

Wavelength is a key factor in determining both the wave speed and the ratio between wave speed and maximum particle speed. The formula for the wave speed \( v = \frac{\omega \lambda}{2\pi} \) clearly ties wavelength to how quickly the wave itself moves through the medium.

Through this investigation, we also see how wavelength appears in the ratio of maximum particle speed to wave speed: \( \frac{2\pi y_m}{\lambda} \).

• Shorter wavelengths means there are more "ups and downs" over a given distance, affecting wave dynamics.
• However, the exercise reveals that material property dependence isn't present in this context, as this ratio involves only amplitude and wavelength, independent of what the wave is traveling through.