A traveling wave is described by the following function y=0.12 cos (4x +2t). Here y(x,t) is the displacement of the particle at location x at time t. All quantities are in SI units. Determine the speed and the direction of the wave.
a. 2 m/s along+x direction.
b. 0.5 m/s along +x direction.
c. 2 m/s along -x direction.
d. 0.5 m/s along - direction.
e. 2 m/s along +y direction.
f. 0.5 m/s along -y direction.

Wave: A wave is a disturbance, that travels through a medium and transfers energy from one point to another, without causing any permanent displacement of the medium itself.

The general equation of a traveling wave can be expressed as

y = Acos(2πft-2πx/λ).................................. Equation 1

Where A = amplitude of the wave, f = frequency of the wave, λ = wavelength of the wave, x = linear distance, t = time, π = pie.

From the question,

the equation of the moving wave is

y = 0.12cos(4x+2t) ................................... equation 2

Comparing equation 1 and 2

-2πx/λ = 4x

λ = -2π/4

λ = -2(3.14)/4

λ = -1.57 m.

Also,

2πft = 2t

f = 2t/2πft

f = 1/π

f = 1/3.14

f = 0.3185 Hz.

Recall that

v = λf.......................... Equation 3

Substitute the value of f and λ into equation 3

v = -1.57(0.3185)

v = - 0.5 m/s.

Note: v is negative because - x direction

Hence the right option is d. 0.5 m/s along -x direction