Two traveling sinusoidal waves are described by the wave functions y1 = 4.85 sin [(4.35x − 1270t)] y2 = 4.85 sin [(4.35x − 1270t − 0.250)] where x, y1, and y2 are in meters and t is in seconds. (a) What is the amplitude of the resultant wave function y1 + y2?

Approximately .

Explanation:

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Notice that sine waves and share the same frequency and wavelength. The only distinction between these two waves is the in .

Therefore, the sum would still be a sine wave. The amplitude of could be found without using calculus.

Consider the sum-of-angle identity for sine:

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Compare the expression to . Let and . Apply the sum-of-angle identity of sine to rewrite .

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Therefore, the sum would become:

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Consider: would it be possible to find and that satisfy the following hypothetical equation?

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Simplify this hypothetical equation:

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Apply the sum-of-angle identity of sine to rewrite the left-hand side:

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Compare this expression with the right-hand side. For this hypothetical equation to hold for all real and , the following should be satisfied:

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Consider the Pythagorean identity. For any real number :

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Make use of the Pythagorean identity to solve this system of equations for . Square both sides of both equations:

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Take the sum of these two equations.

Left-hand side:

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Right-hand side:

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Therefore:

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Substitute back to the system to find . However, notice that the exact value of isn't required for finding the amplitude of .

(Side note: one possible value of is radians.)

As long as is a real number, the amplitude of would be equal to the absolute value of .

Therefore, the amplitude of would be:

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