Wave functions and dynamic question

[yjk91]

Homework Statement

The displacement from equilibrium caused by a wave on a string is given by
y(x, t) = (−0.00217 m)sin[(44.4 m−1)x − (728 s−1)t].

i need to find

(b) number of waves in 1 m
waves

(c) number of complete cycles in 1 s
cycles

(d) wavelength


(e) frequency

The Attempt at a Solution

i first thought the wavelenght was 44.4m because of the formula
Acos2pi(x/λ - t/T)

but then i realized that you had to use the other formula : Acos(kx - wt)

so k = 1/44.4m = 2pi / λ

so λ wave length is 278.97m ?? is this right?

then w = 2pif = 1/728s

so f = 2.186 * 10 ^-4 HZ

and with λ and f you can find the Velocity

not sure if i found λ and f right? can you please tell me if i did it right?

b) number of waves in 1 m
i'm guessing you find this by 1/λ

(c) number of complete cycles in 1 s
and you find this by doing 1/ T

i just want to make sure this is right. if the λ and f are right so i can get the rest of the answer

thank you!

 

[ehild]

The displacement from equilibrium caused by a wave on a string is given by
y(x, t) = (−0.00217 m)sin[(44.4 m−1)x − (728 s−1)t].

The units look weird. Instead of m-1 use m^-1 or 1/m. The same for s-1.
So the wave is y(x,t)=(−0.00217 m) sin[(44.4 m⁻¹)x − (728 s⁻¹)t].

k = 1/44.4m = 2pi / λ

No, k=44.4 m^-1. Otherwise your way of solving the problem will be correct.

ehild

 

[yjk91]

No, k=44.4 m^-1. Otherwise your way of solving the problem will be correct.

ehild

so k =44.4m^-1

does that equal to k = 1/44.4m what does m^-1 mean?

 

[ideasrule]

so k =44.4m^-1

does that equal to k = 1/44.4m what does m^-1 mean?

No, because k=1/44.4m is the same as 1/44.4 m^-1, which is 0.0225 m^-1. k is proportional to the inverse of the wavelength and wavelength is measured in m, so you'd expect the units to be m^-1.

 

[rwisz]

Your calculations for wavelength and frequency appear to be correct. To find the number of waves in 1 m, you can use the formula n = L/λ, where n is the number of waves, L is the length (1 m in this case), and λ is the wavelength. In this case, n = 1/278.97 = 0.0036 waves. Similarly, to find the number of complete cycles in 1 s, you can use the formula n = 1/T, where T is the period (1 s in this case). In this case, n = 1/1 = 1 cycle. So your answers for parts (b) and (c) are correct.

For part (d), you are correct in using the formula λ = 2π/k, but your calculation for k is incorrect. The correct value of k is 44.4 m^-1, which you can obtain by dividing 2π by the wavelength (44.4 m). This gives a wavelength of 0.1416 m or 14.16 cm.

For part (e), you can use the formula f = ω/2π, where ω is the angular frequency (728 s^-1 in this case). This gives a frequency of 115.9 Hz.

Overall, your approach and calculations are correct. Just remember to use the correct values for k and ω. Great job!