The Chemistry Maths Book, Second Edition

3.2 Trigonometric functions 71

EXAMPLE 3.7A harmonic wave travelling in the positive xdirection (a plane

wave) is described by the wave function

(or by the equivalent cosine function), and is shown in Figure 3.12 at timet 1 = 10.

The shortest distance between equivalent points on the curve (the period with respect

to changes in x) is the wavelength λ. The speed of propagation of the wave isv 1 = 1 λν,

where νis the frequency, or number of oscillations per unit time, andτ 1 = 112 νis the

(time) period, the time of a complete oscillation. The number Ais the amplitude of

the wave.

0 Exercises 11

EXAMPLE 3.8In classical mechanics, Newton’s second law of motion states that

the force acting on a body is equal to the mass of the body times its acceleration:

f 1 = 1 ma

For the simple harmonic oscillator (see Example 2.20 and Section 12.5), the force is

f 1 = 1 −kxand the acceleration is the second derivative of distance xwith respect to time

(see Chapter 4 on differentiation):

−=kx m

dx

dt

2

φ

λ

() sinxt A ν

x

,= −t













2 π

A

−A

o λ 2 λ

x

φ(x,0)

....................................................

v=λν

..

...

..

...... ... .. ... ... .. ... ..

... ... .. ... ... .. ... ..

......... ......... .....

...

........

.........

.

....... ......... ........ ..

........

.........

....

...........

.........

....

.. ......... ........ ..

....... ......... ........ ..

........

.........

....

.......................

..........................................................................................

λ=wavelength

.

........

......................

........

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

....

.....

......

.....

......

.....

....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

........

...............

........

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.......

.......................

.......

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

....

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.........

..............

........

.......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

....

......

.....

......

.....

....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.......

.......................

.......

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

......

.....

.......

........

.......

Figure 3.12