The proportion of transmitted energy (Pt) for a transverse wave traveling across the boundary between two media with impedances (z_1, z_2) is given by Pt = (2z_2 / (z_1 + z_2))^2.
When a transverse wave travels from one medium to another with different properties, such as density (ρ) and wave speed (c), some of the wave energy is transmitted across the boundary, and some is reflected. The ratio of the transmitted energy to the incident energy is given by the transmission coefficient (T).
For a wave traveling from medium 1 to medium 2, the transmission coefficient (T) is given by:
T = 2z_2 / (z_1 + z_2),
where z_1 and z_2 are the impedances of the two media, defined as z_1 = ρ_1c_1 and z_2 = ρ_2c_2.
The proportion of energy transmitted (Pt) is then the square of the transmission coefficient:
Pt = T^2.
Substituting the expression for T into the equation for Pt, we get:
Pt = (2z_2 / (z_1 + z_2))^2.
This formula gives the proportion of energy of a transverse wave that is transmitted across the join between two strings of differing density (ρ_1, ρ_2) and wave speed (c_1, c_2) in terms of their impedances (z_1, z_2).
Complete question:
Find the proportion of energy of a transverse wave that is transmitted across the join between two strings of differing density (ρ) and wave speed (c). Write your answer in terms of the impedances of the two strings (z_1 = ρ_1c_1 and z_2 = ρ_2c_2).