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In this paper, we have introduced a new $f(R)$ gravity model as an attempt tohave a model with more parametric control, so that the model can be used toexplain the existing problems as well as to explore new directions in physicsof gravity, by properly constraining it with recent observational data. Herebasic aim is to study the properties of Gravitational Waves (GWs) in this newmodel. In $f(R)$ gravity metric formalism, the model shows the existence ofscalar degree of freedom as like other $f(R)$ gravity models. Due to thisreason, there is a scalar mode of polarization of GWs present in the theory.This polarization mode exists in a mixed state, of which one is transversemassless breathing mode with non-vanishing trace and the other is massivelongitudinal mode. The longitudinal mode being massive, travels at speed lessthan the usual tensor modes found in General Relativity (GR). Moreover, for abetter understanding of the model, we have studied the potential and mass ofscalar graviton in both Jordan frame and Einstein frame. This model can passthe solar system tests and can explain primordial and present dark energy.Also, we have put constraints on the model. It is found that the correlationfunction for the third mode of polarization under certain mass scale predictedby the model agrees well with the recent data of Pulsar Timing Arrays. It seemsthat this new model would be useful in dealing with different existing issuesin the areas of astrophysics and cosmology.

A new f(R) Gravity Model and properties of Gravitational Waves in it