A wave has SHM whose period is 4s while another wave which also possesses SHM has its period 3s. If both are combined, then the resultant wave will have the period equal toA)4sB)5sC)12sD)3.43s

Problem Explanation:
We are given two waves, both exhibiting Simple Harmonic Motion (SHM), with the following periods:
• Wave 1: Period T1=4 sT_1 = 4 \, \text{s}
• Wave 2: Period T2=3 sT_2 = 3 \, \text{s}
The question asks for the period of the resultant wave when these two waves are combined.
Key Concepts:
1. Period of SHM: The period of a wave is the time it takes to complete one full oscillation. For a wave in SHM, the period is related to the angular frequency (ω\omega) by the formula:
T=2πωT = \frac{2\pi}{\omega}
where TT is the period and ω\omega is the angular frequency.
2. Combination of SHM Waves: When two SHM waves of different frequencies are combined (superposed), the resultant wave does not necessarily have a period equal to the simple arithmetic average of the two periods. However, it can be analyzed using the principle of superposition, which may result in interference effects such as constructive or destructive interference.
3. Resultant Period for Two SHM Waves: When combining two SHM waves with different periods, the resultant wave typically exhibits a period corresponding to a combination of the two original periods. If the two waves have periods T1T_1 and T2T_2, the period of the resultant wave is generally the least common multiple (LCM) of T1T_1 and T2T_2.
Step-by-Step Solution:
1. Periods of the individual waves:
o The period of wave 1, T1=4 sT_1 = 4 \, \text{s},
o The period of wave 2, T2=3 sT_2 = 3 \, \text{s}.
2. Finding the least common multiple (LCM): The period of the resultant wave will correspond to the LCM of T1T_1 and T2T_2.
The LCM of 4 and 3 is 1212 (since 12 is the smallest number that is a multiple of both 4 and 3).
Therefore, the period of the resultant wave is Tresultant=12 sT_{\text{resultant}} = 12 \, \text{s}.
Conclusion:
The period of the resultant wave when both waves are combined is 12 seconds. However, none of the options directly correspond to this, but we should also consider the possibility of a misunderstanding or approximations based on possible harmonics or resonance effects. Given the problem as presented, the correct period of the resultant wave is:
12 s(though the given choices are unusual)\boxed{12 \, \text{s}} \quad \text{(though the given choices are unusual)}