# Signal strength of a wave packet

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1. Nov 22, 2016

### Elvis 123456789

1. The problem statement, all variables and given/known data
Assume a wave packet is has contributions from various frequencies, give by g(ω)=C for |ω|<ω0, and g(ω) =0 for elsewhere.

a)What is the signal strength as a function of time, i.e., V(t)=?

b) Sketch g(ω) and V(t); You can use fooplots.com, for example, or python.

c) Indicate Δω and Δt in the above plots; Does the products of these two satisfy ΔωΔt>1/2?

2. Relevant equations
V(t) = 1/√(2π) * integral from -∞ to +∞ of [ g(ω)*exp(iωt)] dω

exp(iωt) = cos(ωt) + isin(ωt)
3. The attempt at a solution

a.) V(t) = 1/√(2π) * integral from -ω0 to +ω0 of [ C*exp(iωt)] dω

using eulers formula and the properties of even and odd functions

V(t) = 2/√(2π) * integral from 0 to +ω0 of [ C*cos(ωt)] dω

V(t) = 2C/√(2π) * sin(ω0t)/t = √(2/π)*ω0C * sin(ω0t)/ω0t

b.) the sketches for g(ω) and V(t) are in the attachments

c.) im not really sure what Δω and Δt in these graphs

#### Attached Files:

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• ###### signal strength vs time.png
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2. Nov 22, 2016

### blue_leaf77

Shouldn't the lower integral limit be $-\omega_0$?
Use the definition of standard deviation. For example for V(t), you will use
$$\Delta t = \sqrt{E[t^2]-(E[t])^2}$$
where $E[\,\,]$ means taking average over the intensity $|V(t)|^2$. Similar arguments for $g(\omega)$.

3. Nov 22, 2016

### Elvis 123456789

I have a factor of 2 in the front to account for that. Is that not right?

4. Nov 22, 2016

### blue_leaf77

Ah sorry I missed that, you are right.