Consider the wave equation:  $y =  10sin(x - 50t)$, What does the number 10 represent?

Analyze the question

The wave equation is y = 10 sin(x − 50t). You are asked what the number 10 stands for.

Explain the concept

The standard form of a travelling wave is:

y = A sin(kx − ωt)

• A is the amplitude, the maximum displacement from the equilibrium position.
• k is the wave number, where k = 2π/λ.
• ω is the angular frequency, where ω = 2πf.

Show the solution

Compare y = 10 sin(x − 50t) with y = A sin(kx − ωt):

• A = 10, which is the number in front of the sine.
• k = 1, so λ = 2π/k = 2π m.
• ω = 50 rad/s.
• Wave speed v = ω/k = 50/1 = 50 m/s.

The sine function only varies between −1 and +1. Multiplying by 10 means y varies between −10 and +10, so the maximum displacement is 10 units.

Check the options

• A. Acceleration: wrong. Acceleration would come from differentiating twice and is not the coefficient.
• B. Speed: wrong. The speed is 50 (from ω/k), not 10.
• C. Amplitude: correct.
• D. Wavelength: wrong. The wavelength is 2π, not 10.

Answer: The number 10 represents the amplitude, which is option C.

Tip: In y = A sin(kx − ωt), the number in front of "sin" is always the amplitude. The number multiplying t, divided by the number multiplying x, gives the speed (v = ω/k).