Question

Two waves travel simultaneously through the same medium. The first wave is described by (y_1(x, t) = (1.00 , {cm}) cos(kx - omega t)) and the second wave by (y_2(x, t) = (8.00 , {cm}) sin(kx - omega t)). What is the amplitude of the resulting wave (in cm)?

Answer 1

The amplitude of the resulting wave, resulting from two waves traveling through the same medium, is found to be approximately 8.06 cm using the principle of superposition and Pythagorean theorem.

Step-by-step explanation:

The two waves traveling simultaneously through the same medium are given by y_1(x, t) = (1.00 cm) cos(kx - ωt) and y_2(x, t) = (8.00 cm) sin(kx - ωt). To find the amplitude of the resulting wave, we use the principle of superposition. However, these two waves are not in the same phase or having the same function (cosine versus sine), so we need to convert one of them to match the other.

To convert the sine function to a cosine function, we use the trigonometric identity sin(θ) = cos(θ - π/2). Therefore,
y_2(x, t) = (8.00 cm) sin(kx - ωt) = (8.00 cm) cos(kx - ωt - π/2). Now, both waves are in the cosine form and can be added easily. To find the amplitude of the resulting wave, we must use the Pythagorean theorem, because the two cosines differ by a phase of π/2.

The amplitude A of the resulting wave is given by A = √(y_1^2 + y_2^2) = √((1.00 cm)^2 + (8.00 cm)^2) = √(1 + 64) = √65 ≈ 8.06 cm. The amplitude of the resulting wave is approximately 8.06 cm.