Particles carry kinetic energy with themselves and transfer energy to other particles in collisions. In this instance, the particle travels for some distance in space and then collides with another particle (Borrowing the terminology from heat and mass transfer we may say there is both mass transfer and energy transfer).
Wave motion is another way of transporting energy. When we say hello to our friend, no material particle is ejected by us from our lips that falls on our friend's ear. We create some disturbance in the part of air close to our lips. Energy is transferred to these air particles by sometimes pushing them ahead or sometimes pulling them back. This affects the density of ther air near our mouth. This disturbance is transferred to the next layer of air and so on till the ear of the listener gets disturbed. Only the disturbance produced in the air travels and the air itself does not move This tope of motion of energy is called a wave motion.
Wave motion on a string.
Consider a long string with one end fixed to a wall and the other held by a person. If the string is held tight, and a small bump is made near the end held by the person, the disturbance travels down the string with a constant speed.
For an elastic and homogeneous string, the bump moves with constant speed to cover equal distances in equal time periods. The shape of the bump is not altered as it moves provided the bump is small.
Equation of a travelling wave
In the string the wave travels from the hand end to the wall end. Let us assue that hand held end is at the left side and the end fixed in the wall is in the right. Hence the X-axis is along the string towards right.
Let function f(t) represent the displacement y of the particle at x = 0 as a function of time
y(x=0,t) = f(t)
As the disturbance is travelling on the string towards right with a constant speed v, the displacement in the y direction produced at the left end at time t, reaches the point x at time t+(x/v)(it takes time equal to x/v). We can express the statement in a different way. The displacement of the particle at point x at time time was originated at the left hand at the time t-(x/v). If the left end is taken as x=0, the displacement at time t-(x/v) will be f(t-x/v) which is y(x=0,t-x/v).
Hence y(x,t) = y(x=0,t-x/v) = f(t-x/v)
y(x,t) represents the displacementof the particle at x at time t. It is generally abbreviated as y and the wave equation is written as
y = f(t-x/v) ...(1)
The equation represents a wave travelling in the positive x direction with a constant speed v.
Function f depends on how the source is providing disturbance.
If the wave is travelling in negative x-direction, with speed v it may be written as
y = f(t+x/v)
Alternative forms of wave equation
y = A sinω(t-x/v)
y = A sin(ωt-kx) …(i)
Where k = ω/v
Remember λ = 2 π v/ ω = 2 π/k or
π/k = λ/2
y = A sin k(vt-x) as kv = ω …(ii)
Interference of waves going in the same direction
Let the amplitudes of the two waves be A1 and A2 and the two waves differ in phase by an angle δ.
Their equations may be written as
y1 = A1 sin(kx-ωt)
y2 = A2 sin(kx-ωt+δ)
By trigonometric properties, the combination of waves (suuperposition of waves) is represented by
= A sin(kx-ωt+ε)
A² = A² cos² ε + A² sin² ε
A1² +A2² +2A1A2cosδ
where A1 +A1cosδ = A cosε
A2sinδ = A sinε
Resultant amplitude is maximum, when cosδ = +1 or δ = 2nπ. The maximum value is A1+A2.
Resultant amplitude is minimum, when cosδ = -1 or cosδ = (2n+1)π . The minimum value is A1-A2.
(These results are used in interference of light.)
Standing waves are produced when two sine waves of equal amplitude and frequency propagate on a long string in opposite directions.
The equations of two waves can be:
y1 = A sin(ωt-kx)
y2 = A sin(ωt+kx)
The equations of the resultant wave will be:
y = y1+y2
= A[sin(ωt-kx) + sin(ωt+kx)]
= 2Asin ωt cos kx
= (2A cos kx) sin ωt
Interpretation of the equation: The amplitude of the wave is |2A cos kx|, but the amplitude is not equal for all the particles.
There are some points where the amplitude |2A cos kx| = 0 all the time.
This will at points where
cos kx = 0
=> kx = (n + ½) π
=> x = (n + ½) λ/2
where n is an integer.
Although these points are not physically clamped, they remain fixed as the two waves pass them simultaneously. These points whose amplitude is zero always are called nodes.
At the points where |cos kx| = 1, the maximum amplitude will be present. These points at which the maximum amplitude is obtained are called antinodes.
When sin ωt = 0, the amplitude of all points is zero which means that all points are at their normal positions or mean positions.
When sin ωt = 1, all points for which cos kx is positive reach their positive maximum displacement. At the same time, all points for which cos kx is negative reach their negative maximum displacement.
From the equation is also clear that the separation between consecutive nodes or consecutive antinodes is λ/2.
As the particles at the nodes do not move at all, energy cannot be transmitted across them.