pat666 said:
Homework Statement
(a) A thin, taut string tied at both ends and oscillating in its third harmonic has its shape
described by the equation
y(x,t) = 5.60cm (sin[(0.0340 rad/cm)x]sin[(50.0 rad/s)t]),
where the origin is at the left end of the string, the x-axis is along the string and the
y-axis is perpendicular to the string.
(i) Draw a sketch that shows the standing wave pattern. (3 marks)
(ii) Find the amplitude of the two traveling waves that make up this standing wave.
(2 marks)
(iii) What is the length of the string? (2 marks)
(iv) Find the wavelength, frequency, period, and speed of the traveling waves. (4 marks)
(v) Find the maximum transverse speed of a point on the string. (2 marks)
(vi)What would be the equation y(x,t) for this string if it were vibrating in its eighth
harmonic? (3 marks)
Homework Equations
The Attempt at a Solution
the only thing i can think to do with this is stick the whole thing in my 89 and do it like that??
I recognise the exam is tommorow so here is what I got from another source.
Solution:
Visualize: What does the third harmonic mode look like ?
y(x,t) = (5.60cm) sin [(0.0340 rad/cm)x] sin [(50.0 rad/s)t]
Compare with standing wave expression
y = 2A sin kx sin ωt
λ3
L
(a) Recall that a standing wave is the sum of two identical traveling waves moving opposite each other. At the anti-nodes, the two waves constructively interfere into a standing wave with an amplitude twice the original amplitude of either traveling wave (we showed this in class!) . In fact,
y (for standing wave) = 2A sin kx sin ωt
Thus, by comparison to the given equation, 2A = 5.6 cm so A = 2.8 cm
(b) From the picture you should note that λ3 = 2L/3 or L = (3/2) λ3
Or you can always get this from the expression for fundamental modes: f3 = 3(v/2L) = v/λ3.
But the wavenumber k gives 2π/λ and examination of the expression above gives k = 0.034 rad/cm = 3.4/m so that λ = 2π/k = 2π/(3.4/m) = 1.85 m. Thus L = (3/2) λ3 = 2.78 m
(c) The wavelength of traveling waves is the same as the standing wave: λ = 1.85 m
(d) The frequency and period of the traveling waves is the same as the standing wave.
Comparison of the sin ωt term with the given expression gives
ω = 50 rad/s or f = ω/2π = 7.96 Hz.
and T = 1/f = 0.1256 sec
(e) The speed of the traveling wave is given by v = λf = (2.78 m)(7.96Hz) = 22.13 m/s
(f) Maximum transverse speed of a point on the string can be found by taking dy/dt and maximixing this.
dy/dt = (5.60 cm)(50 rad/s) sin [(0.0340rad/cm)x] [cos (50 rad/s)t]. The maximum values of sin and cos is 1 so the maximum dy/dt must be (5.6 cm)(50 rad/s) = 280 rad/
