<< Chapter < Page Chapter >> Page >

In this module, we shall consider wave motion resulting from harmonic vibrations. We shall discuss harmonic transverse wave in the context of a string. We assume that there is no loss of energy during transmission of wave along the string. This can be approximated when the string is light and taught. In such condition if we oscillate the free end in harmonic manner, then the vibrations in the string are simple harmonic motion perpendicular to the direction of wave motion. The amplitude of wave form remains intact through its passage along the string

Harmonic wave function

We know that a wave function representing motion in x-direction has the form :

y x , t = A f a x b t

For the case of harmonic vibration, we represent harmonic wave motion in terms of either harmonic sine or cosine function :

y x , t = A sin k x ω t

The argument of trigonometric function “kx-ωt” denotes the phase of the wave function. The phase identifies displacement (disturbance) at a particular position and time. For a particular displacement in y-direction, this quantity is a constant :

k x ω t = constant

where “k” is known as wave number. Its unit is radian/ meter so that the terms in the expression are consistent. Importantly, “k” here is not spring constant as used in the description of SHM.

Amplitude

The amplitude is the maximum displacement on either side of mean position and is a positive quantity. Since sine or cosine trigonometric functions are bounded between “-1” and “1”, the y-displacement is bounded between “-A” and “A”.

Harmonic oscillatory properties

Each particle (or a small segment of string) vibrates in simple harmonic motion. The particle attains the greatest speed at the mean position and reduces to zero at extreme positions. On the other hand, acceleration of the particle is greatest at extreme positions and zero at the mean position. The vibration of particle is represented by a harmonic sine or cosine function. For x=0 :

y x = 0, t = t = A sin - ω t = - A sin ω t

Clearly, the displacement in y-direction is described by the bounded sine or cosine function. The important point here is to realize that oscillatory attributes (like time period, angular and linear frequency) of wave motion is same as that of vibration of a particle in transverse direction. We know that time period in SHM is equal to time taken by the particle to complete one oscillation. It means that displacement of the particle from the mean position at a given position such as x=0 has same value after time period “T” :

y x = 0, t = t = y x = 0, t = t + T

- A sin ω t = - A sin ω t + T = - A sin ω t + ω T

We know that sine of an angle repeats after every “2π” angle. Therefore, the equality given above is valid when :

ω T = 2 π

ω = 2 π T

The linear frequency is given by :

ν = 1 T = ω 2 π

Wavelength and wave number

Wavelength is unique to the study of wave unlike frequency and time period, which is common to oscillatory motion of a particle of the string.

The wavelength is equal to linear distance between repetitions of transverse disturbance or phase. In terms of the speed of the wave, the wavelength of a wave is defined as the linear distance traveled by the wave in a period during which one cycle of transverse vibration is completed. We need to emphasize here that we can determine this wave length by choosing any point on the wave form. Particularly, it need not be a crest or trough which is otherwise a convenient reference point for measuring wavelength. For example, we can consider points A, B and C as shown in the figure. After a time period, T, points marked A’, B’ and C’ are at a linear distance of a wavelength, denoted by “λ”. Clearly, either of linear distances AA’ or BB’ or CC’ represents the wavelength.

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Oscillation and wave motion. OpenStax CNX. Apr 19, 2008 Download for free at http://cnx.org/content/col10493/1.12
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Oscillation and wave motion' conversation and receive update notifications?

Ask