Question

What are progressive waves? Find the equation of progressive wave travelling in +x direction.

Answer 1

Hint: A wave is simply a disturbance propagating through space. It is also interpreted as energy or disturbance propagating energy through a medium. Thus, a wave is a disturbance propagating through space which usually transfers energy. There are two types of waves, stationary and progressive. We shall analyze the progressive wave now.

Complete answer:
In a progressive wave, the particles of the wave undergo a vibratory motion in an elastic medium to pass the disturbance induced from one particle to the successive particle throughout the medium.
The equation of a progressive wave is a cosine function of the position and time of the wave which is expressed as:
$y\left( x,t \right)=A\cos \left( \dfrac{2\pi }{\lambda }x-\dfrac{2\pi }{T}t \right)$
Where,
$y=$ the vertical height of the wave
$x=$ horizontal position of wave at any instant of time
$t=$ any instant of time during wave motion
$A=$ amplitude of wave
$\lambda =$ wavelength of wave
$T=$ time period of wave
The amplitude of the wave is the maximum vertical displacement of the wave from its equilibrium position.The wavelength of wave is the distance between two successive crests or two successive troughs of wave. As the name suggests, the time period of a wave is the time required for a wave to travel one wavelength. The negative sign denotes that the wave is travelling in positive x-direction.

Note:
Our equation of a wave should be an equation for the vertical height of the wave at any horizontal position of the wave with respect to time. Thus, Y should equal as a function of x, it should not be greater than the amplitude of the wave. The path of a wave can be a sine or cosine graph.