Mathematical Representation of a Wave

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A progressive wave can be represented by a sine or cosine curve as shown below.

One radian is 180°/π. A complete revolution is 360° = 2 π radians. The wave is moving from point O, to the right (from left to right →) in the positive x direction as seen in Fig. 1.2.1 below.

The displacement equation given for the particle at O is:

\( \scriptsize y = ASin\omega t \)…. (1)

Let’s say the particle carries on moving along the wave from point O and is now at a point P as shown in Fig. 1.2.2 below. We can say that point O was displaced first before P, that is, we reached point O before we reached P.

From this we can say that:

• Point O is leading P
• or P is lagging behind O

The displacement equation of another particle at P is given by:

\( \scriptsize y = A Sin \left(\omega t \: -\: \phi \right) \)….. (2)

The sign is negative because point O is encountered first before point P. Therefore, point P lags behind point O by angle φ.

For λ length of wave, φ = 2π

For unit length of wave, φ = \( \frac{2 \pi}{\lambda}\)

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