16.2 Mathematics of waves  (Page 3/11)

 Page 3 / 11

Characteristics of a traveling wave on a string

A transverse wave on a taut string is modeled with the wave function

$y\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(kx-wt\right)=0.2\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(6.28\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x-1.57\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t\right).$

Find the amplitude, wavelength, period, and speed of the wave.

Strategy

All these characteristics of the wave can be found from the constants included in the equation or from simple combinations of these constants.

Solution

1. The amplitude, wave number, and angular frequency can be read directly from the wave equation:
$y\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(kx-wt\right)=0.2\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(6.28\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x-1.57\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t\right).$
$\left(A=0.2\phantom{\rule{0.2em}{0ex}}\text{m;}\phantom{\rule{0.2em}{0ex}}k=6.28\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1};\phantom{\rule{0.2em}{0ex}}\omega =1.57\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}\right)$
2. The wave number can be used to find the wavelength:
$\begin{array}{}\\ k=\frac{2\pi }{\lambda }.\hfill \\ \lambda =\frac{2\pi }{k}=\frac{2\pi }{6.28\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}}=1.0\phantom{\rule{0.2em}{0ex}}\text{m}.\hfill \end{array}$
3. The period of the wave can be found using the angular frequency:
$\begin{array}{}\\ \\ \omega =\frac{2\pi }{T}.\hfill \\ T=\frac{2\pi }{\omega }=\frac{2\pi }{1.57\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}}=4\phantom{\rule{0.2em}{0ex}}\text{s}.\hfill \end{array}$
4. The speed of the wave can be found using the wave number and the angular frequency. The direction of the wave can be determined by considering the sign of $kx\mp \omega t$ : A negative sign suggests that the wave is moving in the positive x -direction:
$|v|=\frac{\omega }{k}=\frac{1.57\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}}{6.28\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}}=0.25\phantom{\rule{0.2em}{0ex}}\text{m/s}.$

Significance

All of the characteristics of the wave are contained in the wave function. Note that the wave speed is the speed of the wave in the direction parallel to the motion of the wave. Plotting the height of the medium y versus the position x for two times $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s}$ and $t=0.80\phantom{\rule{0.2em}{0ex}}\text{s}$ can provide a graphical visualization of the wave ( [link] ). A graph of height of the wave y as a function of position x for snapshots of the wave at two times. The dotted line represents the wave at time t = 0.00 s and the solid line represents the wave at t = 0.80 s . Since the wave velocity is constant, the distance the wave travels is the wave velocity times the time interval. The black dots indicate the points used to measure the displacement of the wave. The medium moves up and down, whereas the wave moves to the right.

There is a second velocity to the motion. In this example, the wave is transverse, moving horizontally as the medium oscillates up and down perpendicular to the direction of motion. The graph in [link] shows the motion of the medium at point $x=0.60\phantom{\rule{0.2em}{0ex}}\text{m}$ as a function of time. Notice that the medium of the wave oscillates up and down between $y=+0.20\phantom{\rule{0.2em}{0ex}}\text{m}$ and $y=-0.20\phantom{\rule{0.2em}{0ex}}\text{m}$ every period of 4.0 seconds. A graph of height of the wave y as a function of time t for the position x = 0.6 m . The medium oscillates between y = + 0.20 m and y = −0.20 m every period. The period represented picks two convenient points in the oscillations to measure the period. The period can be measured between any two adjacent points with the same amplitude and the same velocity, ( ∂ y / ∂ t ) . The velocity can be found by looking at the slope tangent to the point on a y -versus- t plot. Notice that at times t = 3.00 s and t = 7.00 s , the heights and the velocities are the same and the period of the oscillation is 4.00 s.

Check Your Understanding The wave function above is derived using a sine function. Can a cosine function be used instead?

Yes, a cosine function is equal to a sine function with a phase shift, and either function can be used in a wave function. Which function is more convenient to use depends on the initial conditions. In [link] , the wave has an initial height of $\text{y}\left(0.00,0.00\right)=0$ and then the wave height increases to the maximum height at the crest. If the initial height at the initial time was equal to the amplitude of the wave $\text{y}\left(0.00,0.00\right)=\text{+}A,$ then it might be more convenient to model the wave with a cosine function.

what is the value of x 6yx7y
what is the formula for frictional force
I believe, correct me if I am wrong, but Ffr=Fn*mu
Grant
frictional force ,mathematically Fforce (Ffr) =K∆R where by K stands for coefficient of friction ,R stands for normal force/reaction NB: R = mass of a body ( m) x Acc.due gravity (g) The formula will hold the meaning if and only if the body is relatively moving with zero angle (∅ = 0°C)
Boay
What is concept associated with linear motion
what causes friction?
Elijah
uneven surfaces cause friction Elijah
Shii
rough surfacea
Grant
what will happen to vapor pressure when you add solute to a solution?
how is freezing point depression different from boiling point elevation?
shane
how is the osmotic pressure affect the blood serum?
shane
what is the example of colligative properties that seen in everyday living?
shane
What is motion
moving place to place
change position with respect to surrounding
to which
to where ?
the phenomenon of an object to changes its position with respect to the reference point with passage of time then it is called as motion
Shubham
it's just a change in position
festus
reference point -it is a fixed point respect to which can say that a object is at rest or motion
Shubham
yes
Shubham
A change in position
Lily
change in position depending on time
bassey
Is there any calculation for line integral in scalar feild?
yes I'm available
Mharsheeraz
what is thrust
when an object is immersed in liquid, it experiences an upward force which is called as upthrust.
Phanindra
@Phanindra Thapa No, that is buoyancy that you're talking about...
Shii
thrust is simply a push
Shii
it is a force that is exerted by liquid.
Phanindra
what is the difference between upthrust and buoyancy?
misbah
The force exerted by a liquid is called buoyancy. not thrust. there are many different types of thrust and I think you should Google it instead of asking here.
Sharath
hey Kumar, don't discourage somebody like that. I think this conversation is all about discussion...remember that the more we discuss the more we know...
festus
thrust is an upward force acting on an object immersed in a liquid.
festus
uptrust and buoyancy are the same
akanbi
the question isn't asking about up thrust. he simply asked what is thrust
Shii
a Thrust is simply a push
Shii
the perpendicular force applied on the body
Shubham
thrust is a force of depression while
bassey
what is friction?
MFON
while upthrust is a force that act on a body when it is fully or partially submerged in a liquid
bassey
mathematically upthrust (u) = Real weight (wr) - Apparent weight (wa) u = wr- wa.
Boay
friction is a force which opposes relative motion.
Boay
how did astromers neasure the mass of earth and sun
wats the simplest and shortest formula to calc. for order of magnitude
papillas
Distinguish between steamline and turbulent flow with at least one example of each
what is newtons first law
It state that an object in rest will continue to remain in rest or an object in motion will continue to remain in motion except resultant(unbalanced force) force act on it
Gerald
Thanks Gerald Fokumla
Theodore
Gerald
it states that a body remains in its state of rest or uniform motion unless acted upon by resultant external force.
festus
it that a body continues to be in a state of rest or in straight line in a motion unless there is an external force acting on it
Usman
state's that a body will continue to maintain it present state of or of uniform unless it's being called upon by an external force
bassey
derive the relation above
formula for find angular velocity
w=v^2/r
Eric
v=wr>2
bassey
Why satellites don't fall on earth? Reason?
because space doesn't have gravity
Evelyn
satellites technically fall to earth but they travel parallel to earth so fast that they orbit instead if falling(plus the gravity is also weaker in the orbit). its a circular motion where the centripetal force is the weight due to gravity
Kameyama
Exactly everyone what is gravity?
the force that attrats a body towards the center of earth,or towards any other physical body having mass
hina
That force which attracts or pulls two objects to each other. A body having mass has gravitational pull. If the object is bigger in mass then it's gravitational pull would be stronger.For Example earth have gravitational pull on other objects that is why we are pulled by earth.
Abdur
Gravity is the force that act on a on body to the center of the earth.
Aguenim
gravity is a force of Attraction
bassey
Qn1(a) Why during the day sky seen blue colour? (b)why during the sunset its seen reddish colour ?
Boay
How the atmosphere reacts with the light from the sun
Grant
what are the application of 2nd law
It's applicable when determining the amount of force needed to make a body to move or to stop a moving body
festus  By Anonymous User        