a8, general physics experiment i spring semester, 2021

A8, General Physics Experiment I () () () Spring Semester, 2022

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Introduction

1. Standing Waves on A String

Goals

� Understand the transverse wave.

� Understand the principle of superposition forwaves.

� Understand the standing wave on a string.

� Estimate the wave speed on the string.

Figure 1: Standing wave patterns on a string: the firstharmonic (top), the second harmonic (middle), and thethird harmonic (bottom).

Figure 2: A sonometer consisting of a massive stringwith a tension and a driver that vibrates the string atone end. The other end of the string is fixed. At certainfrequencies of the vibration, standing waves emerge.

Theoretical BackgroundsConsider a transverse vibration along the y axis ofa string stretched along the x (longitudinal) axis.For simplicity, we restrict ourselves to the vibrationalong the y axis and neglect that along the z axiswhich is actually allowed in nature. We also ignorethe vibration along the longitudinal direction.

(a) A wave is called transverse if the vibrationis perpendicular to the longitudinal directionalong which the wave propagates.

(b) A massive string of the linear mass density ρwith a tension τ is a carrier of a propagatingwave.

(c) The speed of the wave propagation along a vi-brating string

v =

√τ

ρ(1)

was discovered in the late 1500s by VincenzoGalilei, the father of Galileo Galilei.

(d) A transverse wave is described by a time-dependent field y(t, x), a function of both x andt. Here, y(t, x) is the transverse displacementof a tiny segment of the string at x.

(e) A wave propagating along the x axis with thepropagating speed v is a solution to the waveequation:

1

v2∂2

∂t2y(t, x) =

∂2

∂x2y(t, x). (2)

(f) For any differentiable function f(x), y(t, x) =f(x±vt) is a wave satisfying the wave equation(2). Here, f(x − vt) propagates along +i andf(x+ vt) along −i.

(g) A sinusoidal function

y±(t, x) = ym sin(kx∓ ωt+ ϕ) (3)

is a wave that propagates along ±i with thespeed

v =ω

k.

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Here, ϕ is a constant phase, ω is the angularfrequency and k is the wave number:

ω =2π

T,

k =2π

λ,

where T and λ are the period and the wave-length, respectively.

(h) In summary, the wave speed can be expressedas

v =ω

k=

λ

T= λf,

where f = 1/T is the frequency.

(i) A postulate of the wave is that the resultantwave of any two waves is just the algebraic sum:

y(t, x) = y1(t, x) + y2(t, x).

(j) We consider the resultant wave of y+(t, x) andy−(t, x) that are defined in (3). These waves areof common amplitude and phase except thattheir directions of propagation are opposite:

y(t, x) = y+(t, x) + y−(t, x)

= ym[sin(kx− ωt+ ϕ) + sin(kx+ ωt+ ϕ)].

= [2ym sin(kx+ ϕ)] cosωt. (4)

(k) The resultant wave in (4) does not propagatebecause the x dependence and the t dependenceare factorized (decoupled) completely. Thusthe wave does not propagate but stands. Thuswe call it a standing wave.

(l) The x dependence sin(kx + ϕ) has zeros peri-odically at

xnode =nπ − ϕ

k=

(n

2− ϕ

2π

)λ.

We call the points at which the standing wavealways vanishes the nodes. The distance be-tween the two nearest nodes is λ/2.

(m) The x dependence sin(kx + ϕ) has the maxi-mum value 1 periodically at

xantinode =(n+ 1

2)π − ϕ

k=

(n+ 1

2

2− ϕ

2π

)λ.

We call the points at which the standing wavealways has the largest vibration the antin-odes. The distance between the two nearestantinodes is λ/2.

(n) For a stretched string of length L with bothends fixed, the two ends must be nodes simul-taneously. Therefore, the conditions for thestanding waves are

L =λ

2n, n = 1, 2, · · · .

(o) Thus the driver of a sonometer must have thevibrating frequencies satisfying λn = 2L/n tohave standing waves emerge. Thus the wave-lengths and frequencies are quantized as

fn =v

λn= n

v

2L=

n

2L

√τ

ρ,

where the harmonic number n is a positive in-teger and λn is the wavelength of the waveat the nth harmonic. We call the lowest fre-quency (longest wavelength) of n = 1 state thefundamental harmonic or the zeroth har-monic. We call the higher frequency (shorterwavelength) than that of the fundamental har-monic the overtones. For example, n = 2 isthe first overtone or the first harmonic.

Experimental Procedure

1. E1

(a) Connect a PASCO Interface 850 device to acomputer.

(b) Connect a detector to ANALOG INPUT A ofthe interface.



(c) Select Sound Sensor at Hardware Setup .

(d) Connect a driver to OUTPUT 1 of the inter-face.

(e) Apply the AC to the driver by

Signal Generator .

(f) Set the Scope to measure V (t).

(g) Set Fast Monitor Mode and click Monitor tomeasure the intensity.

(h) Increase the frequency applied to the driver and

find the lowest frequency to maximize the am-plitude.

� Note that the resonant frequency of thestring is twice the frequency applied to thedriver.

(i) Repeat the step (h) after replacing the lengthor the density of the string.

2. E2

(a) Connect a String Vibrator to OUTPUT 1 ofthe interface.

(b) Connect a Force Sensor to PASPORT 1 ofthe interface.

� The PASCO Interface 850 will automati-

cally recognize the Force Sensor .

(c) Set the Scope to measure F (t).

(d) Apply the AC to the String Vibrator by

Signal Generator .



(e) Set Fast Monitor Mode and click Monitor tomeasure the tension of the string.

(f) Increase the frequency applied to the

String Vibrator and find the second, third andforth lowest frequency to maximize the ampli-tude.

� Skip the frequency when the end of theString Vibrator operates as an antinode.

(g) Initialize a graph of the horizontal and verticalaxes with t and F (t), respectively.

(h) Make linear fits with a fixed slope m = 0 to theF (t) data to figure out the tension of the string.

(i) Replace the the tension of the string and findthe second lowest frequency to maximize theamplitude.

(j) Repeat the step (g)-(h).



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Discussion (7 points)

Problem 1Problem 2



Problem 3


a8, general physics experiment i spring semester, 2021

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