Mathematics of Traveling Waves

[tburke2]

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.

 

[Dick]

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.

 

[ehild]

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.

 

[ehild]

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.

 

[Dick]

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.