Waves on a String

Even if you have not had calculus, it is worth knowing the language of calculus. (Skip the next three paragraphs if you know about derivatives, partial derivatives, and integrals.)

You have seen delta (D ) used for "change of". A kid's height h increases by 4 inches in a year, and we say D h = 4 in. Assuming a constant growth rate, we could easily calculate the growth in a week (4 in/52) or a second or a microsecond or whatever. Then we use d instead of delta to mean very small change of. The growth rate is dh/dt. A horizontal (x) string may have oscillations in the y direction, and now we need to deal with three variables: x, y, and t. In this case, it is useful to fix your attention on one particular x point, then instead of vertical velocity dy/dt, we call it
y/ t, which means that the other variable (x) is constant. Similarly, we can take a photograph of the string to look at it at a particular time (t constant). Then if you move along the picture of the string a small horizontal distance x and you find that the y value changes a little, the ratio is the slope,
y/ x, a.k.a. the partial of y with respect to x.

Going back to the velocity y/ t, the rate of change of this is the acceleration,
( y/ t)/ t, which is symbolized by 2y/ t2. This is also called the second partial of y with respect to x. Similarly,
the change of y/ x divided by x is
2y/ x2.

While on the subject of the language of calculus, we might as well deal with the integral. This is a sum of a whole bunch of tiny quantities, and instead of S for the sum symbol we use ò . (If this doesn't show up right on your machine, it looks like a tall S that is not as curvaceous.) In the 4 in./year example we say ò dh = 4 in. Think of it as the sum of a whole bunch of little increases of height that occurred all year.

Incidentally, the first calculus I learned was on a lavatory wall. Someone had written
ò d(cabin)/cabin = natural log cabin. Later, someone wrote +C after it. Still later, someone wrote = houseboat. (OK, I know, math humor doesn't make it.) Take calculus sometime and find out about this.

Now consider a horizontal string or hose under tension F. Let m = mass/length, a constant. Then the mass of small length x is m x. The string is given a whack, and we want to find out what happens. Let us apply S fy = may to length x. The right side
is m x 2y/ t2. The left side is the difference between the Fsinq values on the two ends of the length x, where q is the angle from horizontal, and for small angles, sinq is approximately the same as the slope, y/ x. So on the left we have the F times the change of slope. Divide both sides by x, and you have F 2y/ x2 on the left and
m 2y/ t2 on the right. I mentioned in the main waves page that when these two second partials are proportional, the constant of proportionality is the square of the wave speed (more on this below). The units will tell you whether v2 is m /F or F/m . The acceleration is length per time2 and the change of slope/length is 1/length. Show that v2 must be F/m .

By the way, a pure mathematician would prefer death over doing the above stuff. We in physics have more freedom. For example we might let dt stand for a year if we are dealing with some centuries-long process. In math, any amount of time you can specify is too large for dt. To do the above stuff right for a mathematician, do it for D x and take the limit as D x approaches zero, and the result will be the same.

Now why does 2y/ x2 being proportional to
2y/ t2 imply a wave? Before we get to that, here are a few details you need to know: A plane wave traveling in the x direction can be written
y = Asin(kx - w t). A is amplitude, k is angular wave number, radians per distance, and w is angular speed, radians per time. There are 2p radians in a full cycle, so freeze the wave in time and travel an x distance of
one wavelength (l), and you have gone through 2p radians. Hence
k is 2p /l . Similarly at a particular x, in time T (one period), it goes through
2p radians, so w = 2p /T. In period T the wave travels a distance l ,
so v = l /T = (2p /k)/(2p /w ) = w /k. (Frequency f is 1/T, so another handy one is
v = l f.)

Now the main point of all this: We have seen that Newton's 2nd law implies that for low amplitude disturbance on a string with tension F and mass/length m ,
F 2y/ x2=m 2y/ t2 . Check with someone who has had calculus if you haven't, and (s)he will tell you that
if y = Asin(kx-w t), then
2y/ x2 = -k2Asin(kx-w t) and
2y/ t2 = -w 2Asin(kx-w t). Thus
Fk2 = m w 2. And recall that
v = w /k, so show that v2=F/m . So we have shown that a plane sinusoidal wave with velocity (F/m )1/2 satisfies the equation,
F 2y/ x2=m 2y/ t2.

Actually there are other functions that satisfy this equation, but the only other one you need to be concerned with is the standing wave, which mathematically is a combination of two running waves going in opposite directions, and its equation is written
y = Asinkxcosw t. The stationary points are called nodes, and the distance from one node to the next is l /2.

One nifty thing about the use of partial derivatives is that equations just like
F 2y/ x2=m 2y/ t2 govern sound and light (but with different constants).

"You can't go home again," (Thomas Wolfe) but you can go back.

My main pages:

mechanics
fluids, heat, electricity and magnetism
vibrations and waves
quantum
index

Comments, questions: email fredrick.gram @tri-c.edu (remove space before @)