Mathematics of Traveling Waves

In summary: So to find the speed of the particles, we need to differentiate D(x,t) with respect to t and set it equal to 0 to find the maximum and minimum values of D. Then, the speed of the particles at those points would be the time derivative of D.
  • #1
tburke2
6
0
A transverse wave on a cord is represented by D(x,t)=0.22 sin (5.6 x + 34 t)
Determine the velocity of the wave and the minimum and maximum speeds of the particles in the cord.
So velocity can be found with v = f λ
f = ω / 2π and λ = 2π / k
so v = ω / k
= 34 / 5.6
= 6.1

I assume max and min speed of the particles means max and min of |D'| but how do I differentiate a time dependent function like this? I know the min will be at the amplitude and the max at the x intercept.

I've taken multivariable calculus but never had to apply it to a physical system so if someone could point me in the right direction I'd be grateful.
 
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  • #2
tburke2 said:
A transverse wave on a cord is represented by D(x,t)=0.22 sin (5.6 x + 34 t)
Determine the velocity of the wave and the minimum and maximum speeds of the particles in the cord.
So velocity can be found with v = f λ
f = ω / 2π and λ = 2π / k
so v = ω / k
= 34 / 5.6
= 6.1

I assume max and min speed of the particles means max and min of |D'| but how do I differentiate a time dependent function like this? I know the min will be at the amplitude and the max at the x intercept.

I've taken multivariable calculus but never had to apply it to a physical system so if someone could point me in the right direction I'd be grateful.

It doesn't mean max and min of |D'|. It just means solve for the motion of a point where D(x,t)=C where C is some constant. That means 5.6x+34t is a constant B. Differentiate 5.6x+34t=B with respect to t.
 
  • #3
tburke2 said:
A transverse wave on a cord is represented by D(x,t)=0.22 sin (5.6 x + 34 t)
Determine the velocity of the wave and the minimum and maximum speeds of the particles in the cord.
So velocity can be found with v = f λ
f = ω / 2π and λ = 2π / k
so v = ω / k
= 34 / 5.6
= 6.1

I assume max and min speed of the particles means max and min of |D'| but how do I differentiate a time dependent function like this? I know the min will be at the amplitude and the max at the x intercept.

I've taken multivariable calculus but never had to apply it to a physical system so if someone could point me in the right direction I'd be grateful.

It is a transversal wave, D(x,t) means the displacement of a particle at position x (along the line of chord) at time t, and the displacement is perpendicular to the x direction. The velocity of the particle is the time derivative of the displacement. You are right, differentiate D(x,t) with respect to time (so it is partial derivative), and find the maximum and minimum of |D'| . And yes, the speed of the particle is maximum when D=0 and minimum when D is maximum, that is, equal to the amplitude. It does not depend on x.
 
  • #4
Dick said:
It doesn't mean max and min of |D'|. It just means solve for the motion of a point where D(x,t)=C where C is some constant. That means 5.6x+34t is a constant B. Differentiate 5.6x+34t=B with respect to t.
Dick, that is the speed of the wave, not the speed of the oscillating particles of the chord.
 
  • #5
ehild said:
Dick, that is the speed of the wave, not the speed of the oscillating particles of the chord.

Right. Thanks for the correction. Guess I didn't read the full question and was just thinking of wave speed.
 

1. What is a traveling wave in mathematics?

A traveling wave in mathematics is a type of wave that moves through space and time without changing its shape. It is characterized by a constant amplitude and wavelength, and the wave's peak or crest moves continuously in one direction.

2. How is the speed of a traveling wave calculated?

The speed of a traveling wave can be calculated by multiplying the wavelength of the wave by its frequency. This is known as the wave's phase velocity.

3. What is the difference between a transverse and longitudinal traveling wave?

A transverse traveling wave is one in which the particles of the medium move perpendicular to the direction of the wave's propagation, while in a longitudinal traveling wave, the particles move parallel to the direction of the wave's propagation.

4. Can traveling waves interfere with each other?

Yes, traveling waves can interfere with each other, resulting in either constructive interference (when the two waves combine to form a larger wave) or destructive interference (when the two waves cancel each other out).

5. How are traveling waves used in real life applications?

Traveling waves have many real-life applications, such as in communication systems (e.g. radio waves), earthquake detection, and medical imaging techniques like ultrasound. They are also used in studying weather patterns, ocean currents, and other natural phenomena.

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