<< Chapter < Page Chapter >> Page >

The lowest frequency, called the fundamental frequency    , is thus for the longest wavelength, which is seen to be λ 1 = 2 L size 12{λ rSub { size 8{1} } =2`"L"} {} . Therefore, the fundamental frequency is f 1 = v w / λ 1 = v w / 2 L size 12{f rSub { size 8{1} } =v rSub { size 8{w} } /λ rSub { size 8{1} } =v rSub { size 8{w} } /2`"L"} {} . In this case, the overtones    or harmonics are multiples of the fundamental frequency. As seen in [link] , the first harmonic can easily be calculated since λ 2 = L size 12{λ rSub { size 8{2} } =L} {} . Thus, f 2 = v w / λ 2 = v w / 2 L = 2 f 1 size 12{f rSub { size 8{2} } =v rSub { size 8{w} } /λ rSub { size 8{2} } =v rSub { size 8{w} } /2`"L"=2f rSub { size 8{1} } } {} . Similarly, f 3 = 3 f 1 size 12{f rSub { size 8{3} } =3f rSub { size 8{1} } } {} , and so on. All of these frequencies can be changed by adjusting the tension in the string. The greater the tension, the greater v w size 12{v rSub { size 8{w} } } {} is and the higher the frequencies. This observation is familiar to anyone who has ever observed a string instrument being tuned. We will see in later chapters that standing waves are crucial to many resonance phenomena, such as in sounding boxes on string instruments.

The graph shows a wave with wavelength lambda one equal to L, which has two loops. There three nodes and two antinodes in the figure. The length of one loop is L.
The figure shows a string oscillating at its fundamental frequency.
first overtone is shown as the wave length if lambda two is L and there are three nodes and two antinodes in the figure. For first overtone the frequency f two is equal to two times f one.
First and second harmonic frequencies are shown.

Beats

Striking two adjacent keys on a piano produces a warbling combination usually considered to be unpleasant. The superposition of two waves of similar but not identical frequencies is the culprit. Another example is often noticeable in jet aircraft, particularly the two-engine variety, while taxiing. The combined sound of the engines goes up and down in loudness. This varying loudness happens because the sound waves have similar but not identical frequencies. The discordant warbling of the piano and the fluctuating loudness of the jet engine noise are both due to alternately constructive and destructive interference as the two waves go in and out of phase. [link] illustrates this graphically.

The graph shows the superimposition of two similar but non-identical waves. Beats are produced by alternating destructive and constructive waves with equal amplitude but different frequencies. The resultant wave is the one with rising and falling amplitude over different intervals of time.
Beats are produced by the superposition of two waves of slightly different frequencies but identical amplitudes. The waves alternate in time between constructive interference and destructive interference, giving the resulting wave a time-varying amplitude.

The wave resulting from the superposition of two similar-frequency waves has a frequency that is the average of the two. This wave fluctuates in amplitude, or beats , with a frequency called the beat frequency    . We can determine the beat frequency by adding two waves together mathematically. Note that a wave can be represented at one point in space as

x = X cos t T = X cos ft , size 12{x=X" cos"` left ( { {2π t} over {T} } right )=X" cos " left (2π ital "ft" right )","} {}

where f = 1 / T size 12{f= {1} slash {T} } {} is the frequency of the wave. Adding two waves that have different frequencies but identical amplitudes produces a resultant

x = x 1 + x 2 . size 12{x=x rSub { size 8{1} } +x rSub { size 8{2} } "."} {}

More specifically,

x = X cos f 1 t + X cos f 2 t . size 12{x=X"cos" left (2π`f rSub { size 8{1} } t right )+X"cos" left (2π`f rSub { size 8{2} } t right )"."} {}

Using a trigonometric identity, it can be shown that

x = 2 X cos π f B t cos f ave t , size 12{x=2X"cos" left (π`f rSub { size 8{B} } t right )"cos" left (2π`f rSub { size 8{"ave"} } t right )","} {}

where

f B = f 1 f 2 size 12{f rSub { size 8{B} } = lline f rSub { size 8{1} } - f rSub { size 8{2} } rline } {}

is the beat frequency, and f ave size 12{f rSub { size 8{"ave"} } } {} is the average of f 1 size 12{f rSub { size 8{1} } } {} and f 2 size 12{f rSub { size 8{2} } } {} . These results mean that the resultant wave has twice the amplitude and the average frequency of the two superimposed waves, but it also fluctuates in overall amplitude at the beat frequency f B size 12{f rSub { size 8{"B"} } } {} . The first cosine term in the expression effectively causes the amplitude to go up and down. The second cosine term is the wave with frequency f ave size 12{f rSub { size 8{"ave"} } } {} . This result is valid for all types of waves. However, if it is a sound wave, providing the two frequencies are similar, then what we hear is an average frequency that gets louder and softer (or warbles) at the beat frequency.

Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 8

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, College physics -- hlca 1104. OpenStax CNX. May 18, 2013 Download for free at http://legacy.cnx.org/content/col11525/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics -- hlca 1104' conversation and receive update notifications?

Ask