General equation of a travelling wave and standin waves on a string.

[Nikhil Kumar]

Consider a transverse sinusiodal wave on a sting. Let the wave be traveling in positive x-direction. Let its amplitude be A, wave no. be k and angular frequency be ω then the vertical displacement of any particle at a distance x from the origin and at any time t is given as:

y(x,t) = A sin(ωt-kx) ...(1)

does this have the same meaning as (i.e. direction of motion) as

y(x,t) = A sin(kx-ωt) ...(2)?

In most of the books i referred to, eq. (1) is given as the standard equation. However, by using it, i am not able to derive the equations for standing waves on a string fixed at both the ends.

 

[CompuChip]

Note that y(x, t) in (2) is -1 times y(x, t) from (1), so it is not the same wave (but reflected in the x-axis). Therefore, the equations should be the same, perhaps up to a sign. If you get something completely wrong, you might want to post your derivation so we can take a look.

To find out the direction of motion, here's a thought experiment: just follow a maximum. For example, suppose there is one at t = x = 0. If time increases, does this maximum run to the left or to the right?

 

[Nikhil Kumar]

This is the derivation i found in a book:

"y1(x, t) = a sin (kx – ωt) [wave traveling in the positive direction of x-axis] ...(1)
and

y2(x, t) = a sin (kx + ωt) [wave traveling in the negative direction of x-axis].

The principle of superposition gives, for the
combined wave

y (x, t) = y1(x, t) + y2(x, t)

= a sin (kx – ωt) + a sin (kx + ωt)

= (2a sin kx) cos ωt .....(2)

The wave represented by Eq. (1) does not describe a traveling wave, as the waveform
or the disturbance does not move to either side. Here, the quantity 2a sin kx within the
brackets is the amplitude of oscillation of the element of the string located at the position x.
In a traveling wave, in contrast, the amplitude of the wave is the same for all elements.
Equation (15.37), therefore, represents a standing wave, a wave in which the waveform
does not move.

It is seen that the points of maximum or minimum amplitude stay at one position.
The amplitude is zero for values of kx that give sin kx = 0 . Those values are given by

kx = n π(pi) , for n = 0, 1, 2, 3, ...(3)

Substituting k = 2π/λ in this equation, we get
x = 2π/λ , for n = 0, 1, 2, 3, ... (4)

The positions of zero amplitude are called
nodes. "

Please see the equation (1). The wave going in positive x-direction is given as y=A sin(kx-ωt) and not as y=A sin(ωt-kx). Also if we take vice-versa, the conditions for standing waves are reversed and equation for standing waves are obtained as: y= (2a cos kx) sin ωt


Am i wrong somewhere? Someone help me please..