Two sinusoidal waves, which are identical except for a phase shift, travel along in the same direction. The wave equation of the resultant wave is yR (x, t) = 0.70 m sin⎛ ⎝3.00 m-1 x - 6.28 s-1 t + π/16 rad⎞ ⎠. What are the angular frequency, wave number, amplitude, and phase shift of the individual waves?

two sinusoidal waves, which are identical except for a phase shift, travel along in the same direction. The wave equation of the resultant wave is yR (x, t) = 0.70 m sin⎛ ⎝3.00 m-1 x - 6.28 s-1 t + π/16 rad⎞ ⎠. What are the angular frequency, wave number, amplitude, and phase shift of the individual waves?

ω = 6.28 s - 1,

k = 3.00 m - 1,

φ = π rad,

A R = 2 A cos (φ 2),

A = 0.37 m

Explanation:

y1 (x, t) = A sin (k x - ω t + φ),

y 2 (x, t) = A sin (k x - ω t).

from the principle of superposition which states that when two or more waves combine, there resultant wave is the algebriac sum of the individual waves

y1 (x, t) = A sin (k x - ω t + φ), is generaL form of thw wave eqaution

A=amplitude

k=angular wave number

ω=angular frequency

φ = phase constant

k=2π/lambda

ω=2π/T

yR (x, t) = 0.70 m sin{3.00 m-1 x - 6.28 s-1 t + π/16 rad} ... *

two waves superposed to give the above, assuming they are moving in the + x direction

y1 (x, t) = A sin (k x - ω t + φ), ... 1

y 2 (x, t) = A sin (k x - ω t) ... 2

adding the two equation will give

A sin (k x - ω t + φ) + A sin (k x - ω t) ... 3

A (sin (k x - ω t + φ) + sin (k x - ω t)), ... 4

similar to the following trigonometry identity

sina+sinb=2cos (a-b) / 2sin (a+b) / 2

let a = (k x - ω t

b=k x - ω t + φ)

y (x, t) = 2Acos (φ/2) sin (k x - ω t + φ/2)

k=3m^-1

lambda=2π/k=2.09m

ω=6.28 = T=2π/6.28

T=1s

φ/2=π/16

φ=π/8rad

amplitude

2Acos (φ/2) = 0.70 m

A=0.7/2cos (π/8)

A=0.37 m