Problem 23 - Physics

Question

A wave traveling in the \(+x\) direction has an amplitude of \(0.35 \mathrm{~m},\) a speed of \(5.2 \mathrm{~m} /\) \(\mathrm{s},\) and a frequency of \(14 \mathrm{~Hz}\). Write the equation of the wave in the form given by either Equation 16.3 or 16.4

Step-by-Step Solution

Answer

The wave equation is \( y(x,t) = 0.35 \sin(16.91x - 28\pi t) \).

Step 1: Understand Wave Equation Form

Equation 16.3 or 16.4 generally represents a wave traveling in the +x direction as: \( y(x,t) = A \sin(kx - \omega t) \), where \( A \) is the amplitude, \( k \) is the wave number, and \( \omega \) is the angular frequency.

Step 2: Identify Given Values

From the problem, the amplitude \( A = 0.35 \mathrm{~m} \), speed \( v = 5.2 \mathrm{~m/s} \), and frequency \( f = 14 \mathrm{~Hz} \).

Step 3: Calculate Angular Frequency \( \omega \)

The angular frequency is calculated by the formula \( \omega = 2\pi f \). Therefore, \( \omega = 2\pi \times 14 \mathrm{~Hz} = 28\pi \mathrm{~rad/s} \).

Step 4: Calculate Wave Number \( k \)

The wave number \( k \) is given by the formula \( k = \frac{2\pi}{\lambda} \). Since \( \lambda = \frac{v}{f} \), we have \( \lambda = \frac{5.2}{14} \approx 0.3714 \mathrm{~m} \). Thus, \( k = \frac{2\pi}{0.3714} \approx 16.91 \mathrm{~m^{-1}} \).

Step 5: Write the Wave Equation

Substitute the values into the wave equation form: \( y(x,t) = 0.35 \sin(16.91x - 28\pi t) \).

Key Concepts

Amplitude

The amplitude of a wave is a measure of how "tall" or "strong" the wave is, which indicates the maximum extent of oscillation from its equilibrium position. In simpler terms, it's the distance from the wave's midpoint to its peak or trough. For the wave in our problem, the amplitude is given as 0.35 meters. This means that the highest point of this wave is 0.35 meters above the center line, while the lowest point is 0.35 meters below it.

• Amplitude is crucial in determining the energy carried by the wave. The larger the amplitude, the more energy it conveys.
• It is a constant for a particular wave and doesn't depend on frequency or wave speed.

Understanding amplitude helps one grasp the significance of the wave's intensity and the effect it may have during interactions with other media.

Angular Frequency

Angular frequency, represented by the symbol \( \omega \), is key to describing how quickly something oscillates in a wave. Essentially, it tells us how fast the wave makes a complete cycle. You can calculate angular frequency using the formula \( \omega = 2\pi f \), where \( f \) is the frequency of the wave. For this problem, the angular frequency is \( 28\pi \) rad/s.

• Angular frequency is expressed in radians per second.
• It connects the wave's frequency with its rotational equivalent in mathematics.

Grasping angular frequency allows us to understand the timing and rate of oscillations, which is fundamental in analyzing wave behaviors.

Wave Number

The wave number, denoted as \( k \), tells us how many waves fit into a certain distance or how "tight" the waves are within a space. It is linked with the wavelength \( \lambda \) via the formula \( k = \frac{2\pi}{\lambda} \). In the exercise, once we calculate the wavelength as approximately 0.3714 meters, \( k \) becomes approximately 16.91 m\(^{-1}\).

• A higher wave number means more waves fit into a given length, indicating a shorter wavelength.
• The wave number contributes to the spatial frequency of the wave.

Knowing the wave number is important for understanding the distribution and spacing of waves through a medium.

Wave Speed

Wave speed, symbolized as \( v \), quantifies how fast the wave propagates through the medium. It links frequency and wavelength via the equation \( v = f \lambda \). For our wave, the speed is given as 5.2 meters per second.

• Wave speed indicates how quickly energy is transmitted from one location to another by the wave.
• A constant wave speed in a given medium reflects a consistent medium and source characteristics.

Comprehending wave speed is essential for predicting how long it will take a wave to travel between two points and for interpreting the nature of the medium through which it travels.

Frequency

Frequency, indicated by the symbol \( f \), determines how often the wave completes a full cycle within a second. This is expressed in hertz (Hz), with our wave having a frequency of 14 Hz. Essentially, this indicates that 14 complete cycles of the wave occur every second.

• Higher frequencies mean more cycles per second, usually implying more energy.
• Frequency remains constant for a wave type in the same medium.

Understanding frequency is crucial for applications ranging from sound and light to radio waves, as it directly influences both the behavior and the energy of the wave.