Simple harmonic motion and sinusoidal waves

Let us think of grabbing one end of a rope and snap quickly to create a “pulse.” Then, you will see its movement from your end towards the other end. The next video clip shows an animation of such situation. In this video, your grabbing hand is on the left. Please note that each part of the rope described as a small dot does not travel from left to right. Let us take a close look at the red dot in the center. This dot only moves upwards up until the height of 1 and back to the original height of 0 again.

When the snap of your hand is quick enough, the pulse becomes narrow. With a slow hand motion, the pulse becomes wide. In either case, the speed of wave traveling stays the same. The traveling speed is determined by the characteristics of the medium and is independent from the speed of vibration.

Next, let us move your hand up and down repeatedly to make a simple harmonic motion. The next video clip shows an animation of such condition. Let us take a close look at the red dot at the left end, we can observe that it shows a simple harmonic motion. As a result, will see a sinusoidal wave traveling on the rope. The displacement of each dot of the rope repeats 0 → +1 → -1 → 0. In this case, we express the amplitude of the sinusoidal wave is 1.

The following figure shows a snapshot taken at a certain instance. Form this figure, the distance between the consecutive peaks is denoted as $$\lambda,$$ and it is called the wavelength (its unit is meter, or m).

As mentioned above, the speed of the wave is determined by the medium, whereas the speed of vibration is determined by the source. The speed is defined as distance over time. If the wave has the wavelength of $$\lambda$$ [m] and the period of T [s], the speed of the wave $$v = \lambda / T.$$ Because the frequency f [Hz] is the reciprocal of the period T [s], $$ v = \lambda \times f.$$

Let us think of vibration tuning forks in the air. If the tuning forks vibrates at 100 Hz, the period of the sound wave becomes 0.01 s. The speed of sound in the air is about 340 m/s with the temperature of 14 degrees in centigrade. Finally, the wavelength of this sound is approximately 3.4 m.