# 16.2 Mathematics of waves

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By the end of this section, you will be able to:
• Model a wave, moving with a constant wave velocity, with a mathematical expression
• Calculate the velocity and acceleration of the medium
• Show how the velocity of the medium differs from the wave velocity (propagation velocity)

In the previous section, we described periodic waves by their characteristics of wavelength, period, amplitude, and wave speed of the wave. Waves can also be described by the motion of the particles of the medium through which the waves move. The position of particles of the medium can be mathematically modeled as wave function     s , which can be used to find the position, velocity, and acceleration of the particles of the medium of the wave at any time.

## Pulses

A pulse    can be described as wave consisting of a single disturbance that moves through the medium with a constant amplitude. The pulse moves as a pattern that maintains its shape as it propagates with a constant wave speed. Because the wave speed is constant, the distance the pulse moves in a time $\text{Δ}t$ is equal to $\text{Δ}x=v\text{Δ}t$ ( [link] ).

## Modeling a one-dimensional sinusoidal wave using a wave function

Consider a string kept at a constant tension ${F}_{T}$ where one end is fixed and the free end is oscillated between $y=\text{+}A$ and $y=\text{−}A$ by a mechanical device at a constant frequency. [link] shows snapshots of the wave at an interval of an eighth of a period, beginning after one period $\left(t=T\right).$

Notice that each select point on the string (marked by colored dots) oscillates up and down in simple harmonic motion, between $y=+A$ and $y=\text{−}A,$ with a period T . The wave on the string is sinusoidal and is translating in the positive x -direction as time progresses.

At this point, it is useful to recall from your study of algebra that if f ( x ) is some function, then $f\left(x-d\right)$ is the same function translated in the positive x -direction by a distance d . The function $f\left(x+d\right)$ is the same function translated in the negative x -direction by a distance d . We want to define a wave function that will give the y -position of each segment of the string for every position x along the string for every time t .

Looking at the first snapshot in [link] , the y -position of the string between $x=0$ and $x=\lambda$ can be modeled as a sine function. This wave propagates down the string one wavelength in one period, as seen in the last snapshot. The wave therefore moves with a constant wave speed of $v=\lambda \text{/}T.$

Recall that a sine function is a function of the angle $\theta$ , oscillating between $\text{+}1$ and $-1$ , and repeating every $2\pi$ radians ( [link] ). However, the y -position of the medium, or the wave function, oscillates between $\text{+}A$ and $\text{−}A$ , and repeats every wavelength $\lambda$ .

can i get application of projectile motion?
when firing a cannon
morteza
Ok i got a question I'm not asking how gravity works. I would like to know why gravity works. like why is gravity the way it is. What is the true nature of gravity?
gravity pulls towards a mass...like every object is pulled towards earth
Ashok
One answer for that is exchange particles. Those particlescreate a field and if there is a field e.g. electromagnetic there is also force. same should go for gravitational field where exchange particle is called graviton and it creates gravitational field, which acts on everything that has mass.
Resid
@Ashok no that's how gravity works not why it works
Daniel
An automobile traveling with an initial velocity of 25m/s is accelerated to 35m/s in 6s,the wheel of the automobile is 80cm in diameter. find * The angular acceleration
what is the formula for pressure?
force/area
Kidus
force is newtom
Kidus
and area is meter squared
Kidus
so in SI units pressure is N/m^2
Kidus
In customary United States units pressure is lb/in^2. pound per square inch
Kidus
who is Newton?
scientist
Jeevan
a scientist
Peter
that discovered law of motion
Peter
ok
John
but who is Isaac newton?
John
a postmodernist would say that he did not discover them, he made them up and they're not actually a reality in itself, but a mere construct by which we decided to observe the word around us
elo
how?
Qhoshe
Besides his work on universal gravitation (gravity), Newton developed the 3 laws of motion which form the basic principles of modern physics. His discovery of calculus led the way to more powerful methods of solving mathematical problems. His work in optics included the study of white light and
Daniel
and the color spectrum
Daniel
what is a scalar quantity
scalar: are quantity have numerical value
muslim
is that a better way in defining scalar quantity
Peter
thanks
muslim
quantity that has magnitude but no direction
Peter
upward force and downward force lift
upward force and downward force on lift
hi
Etini
hi
elo
hy
Xander
Hello
Jux_dob
hi
Peter
Helo
Tobi
Daniel
what's the answer? I can't get it
what is the question again?
Sallieu
What's this conversation?
Zareen
what is catenation? and give examples
sununu
How many kilometres in 1 mile
Nessy
1.609km in 1mile
Faqir
what's the si unit of impulse
The Newton second (N•s)
Ethan
what is the s. I unit of current
Amphere(A)
imam
thanks man
Roland
u r welcome
imam
the velocity of a boat related to water is 3i+4j and that of water related to earth is i-3j. what is the velocity of the boat relative to earth.If unit vector i and j represent 1km/hour east and north respectively
what is head to tail rule?
kinza
what is the guess theorem
viva question and answer on practical youngs modulus by streching
send me vvi que
rupesh
a car can cover a distance of 522km on 36 Liter's of petrol, how far can it travel on 14 liter of petrol.
Isaac
yoo the ans is 193
Joseph
whats a two dimensional force
what are two dimensional force?