# Can two normal 1D waves form a wave packet?

I have a confusion

A wave packet is described by the superposition of two wave functions: $$Ψ_1(x,t)=A\sin(k_1x−ω_1t)$$ and $$Ψ_2(x,t)=A\sin(k_2x−ω_2t),$$ where $$k_1=2.0×10^6\text{m}^{−1}$$, $$k_2=3.0×10^6\text{m}^{−1}$$, $$ω_1=1.5×10^{14}\text{s}^{−1}$$, and $$ω_2=2.5×10^{14}\text{s}^{−1}$$. Determine the group velocity of the resulting wave packet.

Can just two Normal waves form a wave packet since The book I am following is Beizer Modern Physics show this derivation for group velocity shows just two waves can superimpose to form the wave packet and its group velocity will be equal to $$\frac{\Delta\omega}{\Delta k}$$.

## 2 Answers

The resulting wave will look like that pictured below, and I agree with mmesser314's answer that it would be unusual to call such a wave a "wave packet". That said, it does make sense to speak of its group velocity, and it can be calculated in the way you suggest. It has a meaningful physical interpretation as the velocity of the envelope of the beating wave.

Here, the red point riding on a crest of the "carrier wave" moves at the phase velocity, and the green points on the zeros of the envelope move at the group velocity.

It sounds the problem is misstated a bit. A wave packet is typically a single (perhaps complicated) pulse, where a wave is periodic.

The sum of two periodic waves is periodic. So perhaps it should have said "A wave is described by..."

It takes an infinite number of periodic waves to make a true wave packet. If you choose carefully, you can make a periodic waveform where the period approaches infinity as you add more terms.