By Supawat Poonjiradejma

Date: March 8th, 2015

PHYSICS 101 LEARNING OBJECTIVE

STANDING WAVE

WHAT IS A STANDING WAVE?

• When two harmonic waves of equal amplitude,

wavelength, frequency are moving in opposite direction

WHAT IS A STANDING WAVE?

Node = no motion at 0 amplitude

Antinode = maximum possible amplitude of 2

The node and antinode are λ/4 far apart.

λ/4

KEY EQUATIONS

• D(x,t) = 2Asin(kx)cos(ωt)

• Location of node

• When sin(2πx/λ) = 0, x = nπ

• Location of antinode

• When sin(2πx/λ) = ±1, x = (n+1/2) λ/2

KEY EQUATIONS

• Standing Wave on a string

• Fixed end of string -> length (L) = x

• λ = 2L/n where n = number of antinode in integers

• f = v/λ = nv/2L = n/2L * √T/μ

• fn = nf1 where f1 = fundamental frequency of a string

• Application of standing wave on instruments

• Both end closed/opened

• λ = 2L/n

• f = nv/2L = n/2L * √T/μ

• One end open and another one closed

• λ = 4L/n

• f = nv/4L = n/4L * √T/μ

KEY EQUATIONS

QUESTION 1• A guitar string oscillates in a standing wave pattern after a

person plays with it.

• D(x,t) = 1.5sin(0.6x)cos(20πt)

• 1. What is the wavelength and the frequency?

• 2. What is the distance between two consecutive nodes?

SOLUTION1. The wavelength (λ)

• D(x,t) = 1.5sin(6x)cos(30πt)

• D(x,t) = 2Asin(kx)cos(ωt)

• k = 2π/λ, λ= 2π/k = 2π/6 = 1.05 m

The frequency (f)

ω = 2πf , f = ω/2π = 30π/2π = 15 Hz

2. The distance between two consecutive nodes

• λ = 2L/n where n = 1

• When there is one antinode, there are two nodes at both

ends

• The distance between two nodes is L.

• L = λ/2 = 1.05 m / 2 = 0.525 m

SOLUTION

Nodes

QUESTION 2

• The guitar string has the length of 1.5m long fixed at both ends.

• It has a frequency of 300 Hz. The tension is increased by 10%.

• What is the new fundamental frequency of the string.

SOLUTION

• f = nv/2L = n/2L * √T/μ

• When the tension is increased by 10%, the T will increase to

1.1T

• f = n/2L * √T/μ = ½(1.5m)*√T/μ = 0.75√T/μ = 300 Hz

• f = 0.75√1.1*T/μ = 0.75√T/μ * √1.1 = 300 * √1.1 = 315 Hz

THANKS FOR WATCHING !!!