Power of Transverse Wave: Derive Negative Sign

[jonny001]

As we know Power for transverse wave is P=(Fu)1/2 x A2w2 Sin2(kx-wt) for the wave traveling in +x direction represented by Y=ACos(kx - wt) . However for wave traveling in -x direction is P= -(Fu)1/2 x A2w2 Sin2(Kx-wt) . (Note the negative sign)

The Problem is I am not able to derive this. In derivation I am not getting the negative sign (which i suppose to). Could someone please explain what is power in transverse wave traveling to left represented by Y=ACos(kx + wt).

The concept I am applying is Instantaneous power P=F.V. Please see the diagram.

I hope you can understand what i am saying

 

[LightHero]

The power for a transverse wave travelling to the left is given by the equation P = -(Fu)1/2 x A2w2 Sin2(Kx+wt). This equation is derived from the concept of instantaneous power, which states that the instantaneous power at any point in a wave is equal to the product of the force and the velocity of the wave at that point. Since the force in a transverse wave is proportional to the amplitude of the wave, and the velocity is equal to the product of the angular frequency and the wave number of the wave, we can express the power as P = FuA2w2Sin2(Kx+wt). Since the force in a transverse wave travelling to the left is in the opposite direction to the wave vector, the power must be negative. This is why there is a negative sign in the equation.

 

[astrokreng]

Hello,

Thank you for your question. I understand your confusion about the negative sign in the power equation for a transverse wave traveling in the -x direction. Let's break down the derivation to help you understand where the negative sign comes from.

First, let's start with the general equation for instantaneous power for a transverse wave traveling in the +x direction: P = FV. As you mentioned, this is derived from the equation for power in general, which is P = FV cosθ, where θ is the angle between the force and velocity vectors.

Now, in the case of a transverse wave, the force is given by the tension in the string (F = T) and the velocity is given by the wave speed (V = v). Therefore, we can rewrite the equation as P = Tv cosθ. Since the wave is traveling in the +x direction, the angle between the force and velocity vectors is 0°, which means that cosθ = 1. So, the equation becomes P = Tv.

Next, we can substitute the equation for the velocity of a transverse wave in the equation for power: P = T(kA)w cos(kx - wt). Here, k is the wave number, A is the amplitude, w is the angular frequency, and x and t are the position and time variables, respectively.

If we simplify this equation, we get P = TkwA cos(kx - wt). Now, we can use the trigonometric identity cos(a-b) = cosacosb + sinasinb to rewrite this equation as P = TkwA (coskxcoswt + sinkxsinwt).

Finally, we can substitute the equation for the wave displacement (Y = Acos(kx - wt)) in the equation for power to get P = TkwAY (coskxcoswt + sinkxsinwt). This is the power equation for a transverse wave traveling in the +x direction.

Now, to derive the power equation for a wave traveling in the -x direction, we simply need to substitute the wave displacement equation (Y = Acos(kx + wt)) in the power equation and follow the same steps. When we do this, we get P = -TkwAY (coskxcoswt - sinkxsinwt). As you can see, the only difference between this equation and the one for a wave traveling in the +x direction is the negative