What is the power of a wave in a string?

In summary, the conversation discusses the energy of a small element of a string and its relation to the wave function. The formula for the energy is given, and the book infers that the energy propagates along the string with a velocity of ##v=\Delta x/\Delta t## and the power of the wave is ##P=v[\frac{1}{2}\mu(\frac{\partial y}{\partial t})^2+\frac{1}{2}F(\frac{\partial y}{\partial x})^2]##. However, the speaker has trouble understanding this step because ##x## is not a function of time, and the wave function is defined for all values of ##x## and ##t##. They
  • #1
DavideGenoa
155
5
Hi, friendsi! My text of physics, Gettys', shows how the energy, both kynetic and potential, of a small element ##\Delta x## of a string, through which a wave (whose wave function is ##y:\mathbb{R}^2\to\mathbb{R}##, ##(x,t)\mapsto y(x,t)##) runs, is:

##\Delta E=\Big[ \frac{1}{2}\mu\Big(\frac{\partial y}{\partial t}\Big)^2+\frac{1}{2}F\Big(\frac{\partial y}{\partial x}\Big)^2 \Big]\Delta x##​

where ##\mu## is the linear density of the string and ##F## is its tension. Opportune approximations are made to get this result.

By using an explicit notation for the variables, I would say that the formula means

##\Delta E=\Big[ \frac{1}{2}\mu\Big(\frac{\partial y(x_0,t_0)}{\partial t}\Big)^2+\frac{1}{2}F\Big(\frac{\partial y(x_0,t_0)}{\partial x}\Big)^2 \Big](x-x_0)##​
Everything clear to me until here.
Then, from the formula, my book infers that "the energy propagates along the string with velocity ##v=\Delta x/\Delta t##" and "the power of th wave is ##P=(\Delta E/\Delta x)(\Delta x/\Delta t)##" i.e.
##P=v\Big[ \frac{1}{2}\mu\Big(\frac{\partial y}{\partial t}\Big)^2+\frac{1}{2}F\Big(\frac{\partial y}{\partial x}\Big)^2 \Big]##​
but I do not understand this step, because I do not understand what ##\Delta x/\Delta t## really is... I mean: the ##x## in the expression of ##\Delta E## is not a function of time and ##\Delta E## is defined for any choice of ##x##, ##x_0## and ##t_0## in ##\mathbb{R}##, and ##y## is defined on all ##\mathbb{R}^2##, and not only for ##x=vt##, therefore I do not see how we can define ##\Delta x/\Delta t##, which I explicitly write as ##(x-x_0)/(t-t_0)##, as a well defined velocity, since we cannot consider it as ##(x(t)-x(t_0))/(t-t_0)##: ##x## and ##t## can be arbitrarily chosen and ##x## is not a function of ##t##...

Could anybody explain that step to me? I ##\infty##-ly thank you!
 
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  • #2
In the notation ##v = \Delta x/\Delta t##, ##x = x(t)## is usually the position of a point of fixed phase in the traveling wave, e.g., a wave crest. That might help you to interpret the rest of the notation...
 
  • #3
Thank you very much, oliversum! The problem is that ##y(x,t)##, a wave function, is defined on all ##\mathbb{R}^2##, not only for some ##x=x(t)##: the ##x## in its argument can be any real value independently from ##t##...
 

Related to What is the power of a wave in a string?

1. What is the power of a wave in a string?

The power of a wave in a string refers to the rate at which energy is transferred through the string as a result of the wave. It is measured in watts (W) and depends on the amplitude and frequency of the wave.

2. How is the power of a wave in a string calculated?

The power of a wave in a string can be calculated using the formula P = μω²A²v, where P is power, μ is the mass per unit length of the string, ω is the angular frequency, A is the amplitude of the wave, and v is the speed of the wave.

3. What factors affect the power of a wave in a string?

The power of a wave in a string is affected by the amplitude and frequency of the wave, as well as the mass per unit length and tension of the string. Increasing any of these factors will result in a higher power.

4. Can the power of a wave in a string be negative?

No, the power of a wave in a string cannot be negative. It represents the rate of energy transfer, so it must be a positive value.

5. What is the relationship between power and energy in a wave?

Power and energy are related in that power is the rate at which energy is transferred. The total energy of a wave in a string is equal to the power multiplied by the time the wave is present. So, the higher the power, the more energy is being transferred through the string.

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