Question

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

Answer 1

The amplitude of the resulting wave from the superposition of the two given waves is approximately 8.06 cm.

Step-by-step explanation:

The question deals with the superposition of two waves traveling through the same medium. The first wave is given by y_1(x, t) = (1.00 cm) cos(kx - ωt), and the second wave by y_2(x, t) = (8.00 cm) sin(kx - ωt). To find the resulting wave's amplitude, we need to consider both waves as vectors in the complex plane and calculate the resultant vector's magnitude.

Representing the cosine as a sine with a phase shift, we can rewrite the first wave as y_1(x, t) = (1.00 cm) sin(kx - ωt + π/2). From there, we calculate the resultant amplitude:

A_R = √((1.00 cm)sin(π/2) + (8.00 cm)sin(0))² + ((1.00 cm)cos(π/2) + (8.00 cm)cos(0))²

Plugging in values, we find:

A_R = √(1.00 cm)² + (8.00 cm)² = √(1 + 64) cm = √65 cm ≈ 8.06 cm

The amplitude of the resulting wave is approximately 8.06 cm.