16.2 Mathematics of waves

 Page 1 / 11
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 $\lambda$ .

how can I convert mile to meter per hour
1 mile * 1609m
Boon
hey can someone show me how to solve the - "Hanging from the ceiling over a baby bed ...." question
i wanted to know the steps
Shrushti
sorry shrushti..
Rashid
which question please write it briefly
Asutosh
Olympus Mons on Mars is the largest volcano in the solar system, at a height of 25 km and with a radius of 312 km. If you are standing on the summit, with what initial velocity would you have to fire a projectile from a cannon horizontally to clear the volcano and land on the surface of Mars? Note that Mars has an acceleration of gravity of 3.7 m/s2 .
what is summit
Asutosh
highest point on earth
Ngeh
पृथवी को इसके अक्ष पर कितने कोणीय चाल से घूमाऐ कि भूमधय पे आदमी का भार इसके वासतविक भार से 3/5अधिक हो
best
Murari
At a post office, a parcel that is a 20.0-kg box slides down a ramp inclined at 30.0° 30.0° with the horizontal. The coefficient of kinetic friction between the box and plane is 0.0300. (a) Find the acceleration of the box. (b) Find the velocity of the box as it reaches the end of the plane, if the length of the plane is 2 m and the box starts at rest.
As an IT student must I take physics seriously?
yh
Bernice
hii
Raja
IT came from physics and maths so I don't see why you wouldn't
conditions for pure rolling
Md
the time period of jupiter is 11.6 yrs. how far is jupiter from the sun. distance of earth from rhe sun is 1.5*10 to the power 11 meter.
lists 5 drawing instruments and their uses
that is a question you can find on Google, anyway of top of my head, compass, ruler, protractor, try square, triangles.
Rongfang
A force F is needed to break a copper wire having radius R. The force needed to break a copper wire of radius 2R will be
2F
Jacob
it will be doubled
kelvin
double
Devesh
The difference between vector and scaler quantity
vector has both magnitude & direction but scalar has only magnitude
Manash
my marunong ba dto mag prove ng geometry
ron
how do I find resultant of four forces at a point
Inusah
use the socatoa rule
kingsley
draw force diagram, then work out the direction of force.
Rongfang
In a closed system of forces... Summation of forces in any direction or plane is zero... Resolve if there is a need to then add forces in a particular plane or direction.. Say the x direction... Equate it tk zero
Jacob
define moment of inertia
it is the tendency for a body to continue in motion if is or continue to be at rest if it is.
prince
what is Euler s theorem
what is thermocouple?
joining of two wire of different material forming two junctions. If one is hot and another is cold the it will produce emf...
joining of two metal of different materials to form a junction in one is hot & another is cold
Manash
define dimensional analysis
mathematical derivation?
Hira
explain what Newtonian mechanics is.
a system of mechanics based of Newton laws motion this is easy difenation of mean...
Arzoodan
what is the meaning of single term,mechanics?
jyotirmayee
mechanics is the science related to the behavior of physical bodies when some external force is applied to them
Lalita
SO ASK What is Newtonian mechanics in physics? Newtonian physics, also calledNewtonian or classical mechanics, is the description of mechanical events—those that involve forces acting on matter—using the laws of motion and gravitation formulated in the late seventeenth century by English physicist
Suleiman
can any one send me the best reference book for physics?
Prema
concept of physics by HC verma, Fundamentals of Physics, university of physics
tq u.
Prema
these are the best physics books one can fond both theory and applications.
can any one suggest best book for maths with lot of Tricks?
Vivek
what is the water height in barometer?
SUNEELL
13.5*76 cm. because Mercury is 13.5 times dense than Mercury
LOVE
water is 13.5 times dense than the Mercury
LOVE
plz tell me frnds the best reference book for physics along with the names of authors.
Prema
i recomended the reference book for physics from library University of Dublin or library Trinity college
Arzoodan
A little help here... . 1. Newton's laws of Motion, are they applicable to motions of all speeds? 2.state the speeds which are applicable to Newtons laws of Motion
Derek
mechanics which follows Newtons law
Manash
The definition of axial and polar vector .
Arpita
polar vector which have a starting point or pt. of applications is,force,displacement
jyotirmayee
axial vector represent rotational effect and act along the axis of rotation b
jyotirmayee
prove Newton's first law of motion
prince
Hello frnds what is physics in general?
Ngeh
A block of mass m is attached to a spring with spring constant k and free to slide along a horizontal frictionless surface. At t=0, the block spring system is stretched on amount x>0 from the equilibrium position and is released from rest Vx = 0 What is the period of oscillation of the block? What
Ella
What is the velocity of the block when it first comes back to the equilibrium position?
Ella