Problem 30 - Fundamentals of Physics

Question

Use the wave equation to find the speed of a wave given in terms of the general function \(h(x, t)\) : $$ y(x, t)=(4.00 \mathrm{~mm}) h\left[\left(30 \mathrm{~m}^{-1}\right) x+\left(6.0 \mathrm{~s}^{-1}\right) t\right] . $$

Step-by-Step Solution

Answer

The speed of the wave is 0.2 m/s.

Step 1: Identify the form of the wave equation

The general form of a wave equation is \( y(x, t) = A \cdot h(kx + \omega t) \), where \( A \) is the amplitude, \( k \) is the wave number, and \( \omega \) is the angular frequency.

Step 2: Recognize given parameters

From the equation \( y(x, t) = (4.00 \text{ mm}) h\left[\left(30 \text{ m}^{-1}\right)x + \left(6.0 \text{ s}^{-1}\right) t\right]\), we identify that \( k = 30 \text{ m}^{-1} \) and \( \omega = 6.0 \text{ s}^{-1} \).

Step 3: Calculate wave speed

The speed \( v \) of the wave is given by \( v = \frac{\omega}{k} \). Substitute \( \omega = 6.0 \text{ s}^{-1} \) and \( k = 30 \text{ m}^{-1} \) to find the speed: \[ v = \frac{6.0 \text{ s}^{-1}}{30 \text{ m}^{-1}} = 0.2 \text{ m/s} \].

Key Concepts

Wave Speed

Wave speed is a measure of how fast the wave travels through a medium. Imagine watching waves move across a pond. Just like those ripples, wave speed tells us how quickly the wave crests go from one spot to another.

• In mathematics, the wave speed \( v \) can be calculated using the formula \( v = \frac{\omega}{k} \), where \( \omega \) is the angular frequency and \( k \) is the wave number.
• The wave speed depends on the properties of the wave and the medium through which it is moving.

In the original equation given, \( \omega = 6.0 \text{ s}^{-1} \) and \( k = 30 \text{ m}^{-1} \), we use these values to find that the speed of the wave is \( 0.2 \text{ m/s} \).
Remember, understanding wave speed is key to analyzing and predicting how waves behave in various environments.

Amplitude

Amplitude refers to the maximum extent of a vibration or oscillation, measured from the position of equilibrium. Think of it as the height of a wave crest above its average position.

• It is represented by the symbol \( A \) and tells us how far the wave moves from its resting position.
• In the context of sound, larger amplitudes mean louder sounds. For light, larger amplitudes mean brighter light.

In our wave equation \( y(x, t) = (4.00 \text{ mm}) h(...) \), the amplitude is \( 4.00 \text{ mm} \).
This indicates how high above and below the central value the waves move due to oscillations. Understanding amplitude is crucial for interpreting the energy and intensity of waves in different scenarios.

Wave Number

Wave number \( k \) is like a measure of how many wave crests are packed into a certain distance. It's a little like frequency, but instead of time, it involves space.

• The wave number is defined as \( k = \frac{2\pi}{\lambda} \), where \( \lambda \) is the wavelength.
• A higher wave number means more wave crests in a given length, indicating shorter wavelengths.

For the wave in our equation, \( k = 30 \text{ m}^{-1} \).
This means in just one meter, there are 30 crests of the wave, portraying a short wavelength. Understanding wave number helps in analyzing the spatial distribution of waves.

Angular Frequency

Angular frequency \( \omega \) is a bit like the heartbeat of the wave. It tells us how fast the wave oscillates or how quickly it cycles back to its starting point.

• Angular frequency is connected to the frequency of the wave by the formula \( \omega = 2\pi f \), where \( f \) is the usual frequency measured in cycles per second or Hertz.
• It is measured in radians per second, making it a handy tool for relating rotational and wave motion.

In the given equation, \( \omega = 6.0 \text{ s}^{-1} \).
This tells us that the wave completes 6 cycles every second, making it quite fast. Grasping the concept of angular frequency is essential for exploring the nature and behavior of periodic waves.