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

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

Solution: The speed of the wave v is given by v = fλ. Here, f = 2 Hz and λ = 1 m (from the wave equation). Thus, v = 2 * 1 = 2 m/s.

Steps: 8

Step 1: Identify the wave equation given: y(x, t) = 0.1 sin(2π(0.5x - 2t)).
Step 2: Recognize the general form of a wave equation: 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 * 2.
Step 4: Calculate the wave number k: k = 2π * 0.5 = π.
Step 5: Calculate the angular frequency ω: ω = 2π * 2 = 4π.
Step 6: Find the frequency f using the formula f = ω / (2π): f = 4π / (2π) = 2 Hz.
Step 7: Calculate the wavelength λ using the formula λ = 2π / k: λ = 2π / π = 2 m.
Step 8: Use the wave speed formula v = f * λ: v = 2 Hz * 2 m = 4 m/s.