A longitudinal wave is represented by the equation z(z, t) = −2.0 cm sin(1,200 rad/s t − 20 m⁻¹ z). Another longitudinal wave is represented by the equation z(z, t) = +2.0 cm sin(1,200 rad/s t + 20 m⁻¹ z). What is the equation that represents the superposition of the two waves? A) z(z, t) = -4.0 cm cos(1,200 rad/s t) sin(20 m−1 z) B) z(z, t) = +4.0 cm cos(1,200 rad/s t) sin(20 m⁻¹ z) C) z(z, t) = +4.0 cm cos(1,200 rad/s t) cos(20 m⁻¹ z) D) z(z, t) = −2.0 cm cos(1,200 rad/s t) sin(20 m⁻¹ z)

Question

Quizplus · Accepted Answer

When two waves superpose, the displacement of the resulting wave is the sum of the individual displacements of the two waves at that point. Therefore, the equation representing the superposition of the two waves is: z(z, t) = −2.0 cm sin(1,200 rad/s t − 20 m⁻¹ z) + 2.0 cm sin(1,200 rad/s t + 20 m⁻¹ z) Using the identity sin(A+B) = sin(A)cos(B) + cos(A)sin(B), we can simplify this equation to: z(z, t) = 2.0 cm [sin(1,200 rad/s t)cos(20 m⁻¹ z) - cos(1,200 rad/s t)sin(20 m⁻¹ z)] which simplifies further to: z(z, t) = 2.0 cm cos(1,200 rad/s t) sin(20 m⁻¹ z) - 2.0 cm sin(1,200 rad/s t) cos(20 m⁻¹ z) Thus, the best choice is B, which matches this equation.

A Longitudinal Wave Is Represented by the Equation Z(z, T)