A wave traveling along a string is described by the equation y(x, t) = 0.1 sin(2

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Question: A wave traveling along a string is described by the equation y(x, t) = 0.1 sin(2π(0.5x - 4t)). What is the wave speed?

Options:

Correct Answer: 4 m/s

Solution:

The wave speed v can be found from the equation v = fλ. Here, f = 4 Hz and λ = 2 m (from k = 2π/λ, k = 1 m⁻¹). Thus, v = 4 Hz * 2 m = 8 m/s.

A wave traveling along a string is described by the equation y(x, t) = 0.1 sin(2π(0.5x - 4t)). What is the wave speed?

A wave traveling along a string is described by the equation y(x, t) = 0.1 sin(2π(0.5x - 4t)). What is the wave speed?

Step 1: Identify the wave equation given: y(x, t) = 0.1 sin(2π(0.5x - 4t)).
Step 2: Recognize that the general form of a wave equation is y(x, t) = A sin(kx - ωt), where A is amplitude, k is the wave number, and ω is the angular frequency.
Step 3: From the equation, identify k and ω. Here, k = 2π * 0.5 and ω = 2π * 4.
Step 4: Calculate k: k = 2π * 0.5 = π m⁻¹.
Step 5: Calculate ω: ω = 2π * 4 = 8π rad/s.
Step 6: Find the frequency f using the formula f = ω / (2π). So, f = 8π / (2π) = 4 Hz.
Step 7: Calculate the wavelength λ using the formula k = 2π / λ. Rearranging gives λ = 2π / k = 2π / π = 2 m.
Step 8: Use the wave speed formula v = f * λ. Substitute f = 4 Hz and λ = 2 m into the formula: v = 4 Hz * 2 m.
Step 9: Calculate the wave speed: v = 8 m/s.

Wave Equation Analysis – Understanding the relationship between wave parameters such as frequency, wavelength, and wave speed.
Wave Speed Calculation – Applying the formula v = fλ to determine the speed of a wave from its frequency and wavelength.
Understanding Wave Properties – Recognizing how the wave's mathematical representation relates to physical properties like speed.