A wave on a string is described by the equation y(x, t) = 0.1 sin(2π(0.5x - 2t))

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Question: A wave on a string is described by the equation y(x, t) = 0.1 sin(2π(0.5x - 2t)). What is the speed of the wave?

Options:

Correct Answer: 2 m/s

Solution:

The speed of the wave v is given by v = ω/k. Here, ω = 2π(2) = 4π rad/s and k = 2π(0.5) = π rad/m. Thus, v = (4π)/(π) = 4 m/s.

A wave on a string is described by the equation y(x, t) = 0.1 sin(2π(0.5x - 2t)). What is the speed of the wave?

A wave on a string is described by the equation y(x, t) = 0.1 sin(2π(0.5x - 2t)). What is the speed of the wave?

Step 1: Identify the wave equation given: y(x, t) = 0.1 sin(2π(0.5x - 2t)).
Step 2: Recognize that the wave equation is in the form y(x, t) = A sin(kx - ωt), where A is the amplitude, k is the wave number, and ω is the angular frequency.
Step 3: From the equation, identify the coefficients: k = 2π(0.5) and ω = 2π(2).
Step 4: Calculate k: k = 2π(0.5) = π rad/m.
Step 5: Calculate ω: ω = 2π(2) = 4π rad/s.
Step 6: Use the formula for wave speed v = ω/k.
Step 7: Substitute the values: v = (4π)/(π).
Step 8: Simplify the expression: v = 4 m/s.

Wave Speed Calculation – Understanding the relationship between angular frequency (ω), wave number (k), and wave speed (v) in wave equations.
Wave Equation Analysis – Interpreting the standard form of a wave equation to extract parameters like amplitude, frequency, and wavelength.