Problem 50 - University Physics with Modern Physics

Question

Write the equation for a sinusoidal wave propagating in the negative \(x\) -direction with a speed of \(120 . \mathrm{m} / \mathrm{s}\), if a particle in the medium in which the wave is moving is observed to swing back and forth through a \(6.00-\mathrm{cm}\) range in \(4.00 \mathrm{~s}\). Assume that \(t=0\) is taken to be the instant when the particle is at \(y=0\) and that the particle moves in the positive \(y\) -direction immediately after \(t=0\).

Step-by-Step Solution

Answer

Question: A sinusoidal wave is observed to travel in the negative x-direction with a wave speed of 120 m/s. The oscillating particle is observed to swing back and forth through a 6.00 cm range and take 4.00 seconds to complete an oscillation. Find the equation of the wave. Answer: The equation of the wave is given by y(x,t) = 0.030 * cos(π/240 * x + 0.5π * t - π/2), where x is the position, and t is the time.

Step 1: Find the amplitude of the wave

The amplitude (A) is the maximum displacement of the particles from their mean position. In this problem, we are given that a particle is observed to swing back and forth through a 6.00 cm range. Therefore, the amplitude of the wave is half of this range. A = 6.00 cm / 2 = 3.00 cm = 0.030 m

Step 2: Find the frequency and angular frequency of the wave

The time taken for a complete oscillation of the particle is given as 4.00 seconds. This time is called the time period (T) of the wave. The frequency (f) is the inverse of the time period. f = 1 / T = 1 / 4.00 s = 0.25 Hz The angular frequency (ω) is related to the frequency by the following formula: ω = 2πf ω = 2π(0.25 Hz) = 0.5π rad/s

Step 3: Determine the wave number (k) of the wave

The wave speed (v) is given as 120 m/s. The wave number (k) is related to the wave speed and angular frequency by the following formula: v = ω / k Rearrange the equation and solve for k: k = ω / v = (0.5π rad/s) / (120 m/s) = π/240 rad/m

Step 4: Write the equation for the sinusoidal wave

Since the wave is propagating in the negative x-direction and particles move in the positive y-direction immediately after t=0, the equation for the sinusoidal wave can be represented as: y(x,t) = A * cos(kx + ωt + φ) Here, φ is the phase constant. As we are assuming that at t=0, the particle is at y=0 and moves in the positive y-direction immediately after t=0, this means the wave is at its mean position and hence has a phase constant of -π/2. φ = -π/2 Now, substitute the values of A, k, ω, and φ in the equation: y(x,t) = 0.030 * cos(π/240 * x + 0.5π * t - π/2)

Key Concepts

Wave Propagation

Wave propagation is the movement of waves through a medium. In this case, the wave is a sinusoidal wave moving along a path. The direction of the wave is important in its analysis. Here, the wave travels in the negative x-direction, meaning it moves opposite to the positive x-axis. This direction affects the wave equation and helps determine the signs of different parameters, like the wave number.

• Direction: Negative x-axis indicates opposing travel to positive direction.
• Wave Type: Sinusoidal wave demonstrates cyclic movement.
• Medium: The material through which the wave travels determines the speed.

Understanding wave propagation is essential to describe how energy travels from one point to another in a specific medium. In our example, the speed of propagation is 120 meters per second, illustrating swift movement through the medium.

Wave Amplitude

A wave's amplitude indicates the maximum extent of its oscillation, measured from its mean position. In simpler terms, it represents the height of the wave. For our problem, we are informed that a particle moves in a range of 6.00 cm. This full range needs to be divided by two to get the amplitude.

• Maximum Displacement: Amplitude is half of the total oscillation range.
• Example: With a 6.00 cm total range, the amplitude is 3.00 cm.

Amplitude is crucial since it tells us how significant the movement or oscillation is. Here, converting 3.00 cm to meters gives us an amplitude of 0.030 meters. This value is a vital part of our sinusoidal wave equation, showing the wave's strength or energy level.

Angular Frequency

Angular frequency is a measure of how fast the wave oscillates or cycles back and forth. It relates to the frequency, which is the number of cycles per second. Our problem provides a period of 4.00 seconds, the time it takes for one complete cycle.

• Time Period (T): The interval taken for one full oscillation.
• Frequency (f): Calculated as the inverse of time period, here 0.25 Hz.
• Angular Frequency (ω): Given by the formula ω = 2πf.

By substituting the frequency into the formula, we find that the angular frequency is 0.5π rad/s. This value is critical in understanding the wave's energetic behavior and is used explicitly to define the sinusoidal wave equation.

Phase Constant

The phase constant is an essential parameter in the sinusoidal wave equation. It dictates the initial angle or phase of the wave at time zero. In our problem, it is specified that the particle is at its equilibrium position ( y = 0 ), moving in the positive y -direction, right after ( t = 0 ).

• Wave Starting Point: At ( t = 0 , y = 0 ), the wave is at mean position.
• Direction of Movement: Positive y-direction after time zero.
• Phase Constant (φ): In this scenario, φ = -π/2.

Choosing the correct phase constant is key to accurately modeling the wave's starting point. Thus, the phase constant -π/2 is chosen for our equation to show that the wave aligns with the described initial conditions. This phase alignment ensures the wave is correctly described by the equation.