Why does a wave composed of two waves have an FFT with only one peak?

Question

I am trying to find a value from a given mathematical function whose input is supposed to be derived from FFT components. Essentially, I follow the procedure from [1] which assumes that a given sine wave (regular wave) is composed of a cosine and sine wave of different amplitude but the same frequency.

How can I find the amplitudes of each decomposed wave if the FFT shows just one peak at the wave frequency?

The signal is a regular sine wave superimposed with 2 waves traveling in the opposite direction. We have two singals (i.e. two measurment devices) whose components are required for the mathemtical equation. So,

Signal₁ = Wave₁ + Wave₂ = a₁cos(kx-ft+e₁) + a₂cos(kx+ft+e₂) = A₁cos(ft) + B₁sin(ft)

Signal₂ = Wave₁ + Wave₂ = a₁cos(kx-ft+e₁) + a₂cos(kx+ft+e₂) = A₂cos(ft) + B₂sin(ft) where,

• a = ampltiude
• k = wavenumber
• f = frequency
• e = phase
• t = time

The paper states A₁,A₂ ,B₁ and A₂ can be estimated through Fourier Analysis to be plugged into a equation.

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[1] Yoshimi Goda, Yasumasa Suzuki, "Estimation of Incident and Reflected Waves in Random Wave Experiments", Coastal Engineering 1976 -1977

Answer 1

The generalized form $Ae^{j\theta}$ with $A$ and $\theta$ as real numbers is a phasor with magnitude $K$ and angle $\theta$. A "positive" frequency $\Omega_o$ is such a phasor with constant magnitude $A$ and rotating counter-clockwise with angle that increases linearly with time as $\Omega_o t$. (And similarly a negative frequency would be rotating clock-wise). The real and imaginary components of such a single frequency tone would be given as $\cos(\Omega_o t)$ and $\sin(\Omega_o t)$ respectively which is Euler's formula. Therefore any such "single tone" $Ae^{j\Omega_o t}$ would have cosine and sine components. The Discrete Fourier Transform is composed of such "single tones" with each bin represented in the time domain as $Ae^{j k \omega_o n}$, where $\omega_o = 2\pi/N$.